#6584801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#25115	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1913 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)(13,14,18,21)(15,19,23,27)(16,24,26,28)(17,22,25,20);(1,2)(3,4)(6,10)(11,12)(13,15,18,23)(14,19,21,27)(16,22,26,20)(17,24,25,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,25,19,18,26,17,27)(14,15,22,28,21,23,20,24);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(15,23)(16,26)(19,27)(24,28);(2,4)(15,23)(16,26)(19,27)(24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491997);