#6443700

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250065, #72256998, #72257026
maximal normal subgroup of	#72250065, #72256998, #72257026
maximal quotient of	#72308550

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350896);