#6815167

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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
prime radical	#31057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722363);