#7104310

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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	#59304	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1839 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:

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

Polycyclic presentation:

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

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, 7011506);