#6547141

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(21,22,26,30,27,31,25,35)(23,32,33,36,24,29,34,28);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(23,24)(28,36)(29,32)(33,34);(6,10)(11,17)(12,18)(19,20)(21,23,27,24)(22,28,31,36)(25,33,26,34)(29,35,32,30);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,24,27,23)(22,29,31,32)(25,33,26,34)(28,35,36,30);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,31,35)(23,33,24,34)(28,29,36,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6454337);