#6578051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2097 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)(8,12)(9,10)(13,14)(15,18)(16,24)(17,21)(19,23)(20,25)(22,27)(26,28);(1,2)(3,4)(6,10)(11,12)(13,15,18,14)(16,22,26,20)(17,21,19,23)(24,25,28,27);(2,4)(13,16)(14,20)(15,22)(17,25)(18,26)(19,27)(21,28)(23,24);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,17,18,19)(14,21,15,23)(16,25,26,27)(20,28,22,24);(2,4)(5,8,9,7)(6,12,10,11)

Polycyclic presentation:

2^9|b^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485247);