#6585609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#28231	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 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)(13,14,18,17)(15,22,19,20)(16,24,27,28)(21,23,26,25);(1,2)(3,4)(6,10)(11,12)(13,15,18,19)(14,20,17,22)(16,25,27,23)(21,24,26,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,15,23,18,27,19,25)(14,21,20,28,17,26,22,24);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,17,18,14)(15,22,19,20)(16,26,27,21)(23,24,25,28);(2,4)(5,8,9,7)(6,12,10,11)(13,15,18,19)(14,22,17,20)(16,23,27,25)(21,24,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492805);