#72321130

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂³	C₂²×#97
prime radical	C₂²×#41	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6,9,13,10,12,17,8)(7,11,16,18,15,19,20,14)(21,22,24,31,26,33,25,30,28,35,42,49,27,34,41,32)(23,29,36,45,39,47,38,43,37,44,50,52,40,48,51,46);(5,7)(6,11)(8,14)(9,16)(10,15)(12,19)(13,18)(17,20)(21,23)(22,29)(24,36)(25,38)(26,39)(27,40)(28,37)(30,43)(31,45)(32,46)(33,47)(34,48)(35,44)(41,51)(42,50)(49,52);(5,8,17,12,10,13,9,6)(7,14,20,19,15,18,16,11)(21,24,27,41,28,42,26,25)(22,30,33,49,35,32,34,31)(23,36,40,51,37,50,39,38)(29,43,47,52,44,46,48,45);(2,4)(6,12)(7,15)(8,13)(16,20)(23,37)(29,44)(36,50)(38,51)(39,40)(43,46)(45,52)(47,48)

Polycyclic presentation:

2^10|a^2=e,c^2=egij,e^2=h,g^2=i,h^2=j,i^2=j,c^a=ci,d^a=df,d^b=dhij,d^c=dhi,e^c=ej,g^a=gj,g^c=gj

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321141, #72321306, #72321337, …
maximal normal subgroup of	#72321141, #72321306, #72321337, …
maximal quotient of	#72322682, #72323009

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.5*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);