#6811671

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

2372 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)(7,10)(8,12)(9,11)(14,16)(15,22)(17,19)(20,27)(23,28)(24,26);(2,3)(5,7,10,6)(8,12,9,11)(13,14)(15,20)(16,18)(17,25)(19,21)(22,28)(23,24)(26,27);(1,2,4,3)(13,15)(14,20)(16,23)(17,22)(18,24)(19,26)(21,27)(25,28);(5,8,10,9)(6,11,7,12)(13,16,18,14)(15,23,24,20)(17,21,19,25)(22,27,26,28);(1,3,4,2)(5,9,10,8)(6,11,7,12)(13,17,18,19)(14,21,16,25)(15,22,24,26)(20,27,23,28)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6718867);