#6571581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2998 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)(3,4)(21,22,25,28)(23,27,30,34)(24,26,32,29)(31,33,36,35);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,27)(24,31)(25,30)(26,33)(28,34)(29,35)(32,36);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(22,28)(23,30)(26,29)(31,36);(8,16)(11,18)(14,20)(17,19)(23,30)(27,34)(31,36)(33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei,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, 6478777);