#6598461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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
prime radical	C₂×#3321	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505657);