#72323178

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(7,13)(11,16)(14,19)(18,20)(21,22,24,29,26,31,35,28,27,32,34,36,25,30,23,33);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20);(2,4)(6,10)(11,16)(12,17)(18,20);(5,7,8,14,9,13,15,19)(6,11,12,18,10,16,17,20)(21,23,25,34,27,35,26,24)(22,28,30,29,32,33,31,36)

Polycyclic presentation:

2^10|a^2=e,d^2=g,e^2=h,g^2=i,h^2=j,c^a=cf,c^b=cij,d^a=di

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323188
maximal normal subgroup of	#72323188
maximal quotient of	#72323179

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, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);