#6814581

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,26)(27,28);(2,3)(5,7)(6,9)(8,12)(10,11)(14,19)(15,22)(16,25)(26,27);(1,2,4,3)(13,15,21,14,17,22,18,19)(16,23,27,20,25,28,26,24);(6,10)(9,11)(13,16)(14,20)(15,23)(17,25)(18,26)(19,24)(21,27)(22,28);(1,3,4,2)(7,12)(9,11)(16,25)(20,24)(23,28)(26,27)

Polycyclic presentation:

2^9|c^2=gh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6721777);