#6814468

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6721664);