#6817311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6724507);