#6544092

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

2970 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)(22,27)(23,30)(24,25)(29,32)(31,33)(34,36);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,22)(23,28)(24,27)(25,32)(26,29)(30,34)(31,35)(33,36);(6,10)(11,17)(12,18)(19,20)(21,23)(22,28)(24,30)(25,33)(26,31)(27,34)(29,35)(32,36);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(22,29)(27,32)(28,35)(34,36);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,24,26,25)(22,27,29,32)(23,30,31,33)(28,34,35,36)

Polycyclic presentation:

2^9|a^2=fgi,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6451288);