#6585592

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	#28227	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1897 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)(13,14,18,16)(15,22,25,28)(17,20,26,21)(19,27,23,24);(1,2)(3,4)(6,10)(11,12)(13,14)(15,23)(16,18)(17,21)(19,25)(20,26)(22,27)(24,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,24,18,25,26,27)(14,19,21,22,16,23,20,28);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,18,14)(15,19,25,23)(17,20,26,21)(22,27,28,24);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,26)(14,20,16,21)(15,24,25,27)(19,28,23,22)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6492788);