#72265443

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#778809	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#392 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=i,b^2=f,d^2=h,i^2=j,b^a=bij,d^b=dh,d^c=dhj,e^a=ehij,e^b=ef,e^c=efg,e^d=ef,i^b=ij,i^e=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265530
maximal normal subgroup of	#72265530

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