#6668609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2716 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575805);