#6547165

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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1764 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)(21,22,27,31,26,28,25,32)(23,33,34,29,24,30,35,36);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(22,28)(23,24)(31,32)(34,35);(6,10)(11,17)(12,18)(19,20)(21,23,26,24)(22,29,28,36)(25,34,27,35)(30,31,33,32);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,24,26,23)(22,30,28,33)(25,34,27,35)(29,31,36,32);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,26,27)(22,31,28,32)(23,34,24,35)(29,33,36,30)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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, 6454361);