#7165986

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,25,29)(23,31,24,27)(26,28,34,30)(32,35,33,36);(1,2)(3,4)(7,13)(8,16)(10,17)(11,15)(21,23)(22,27)(24,25)(26,32)(28,36)(29,31)(30,35)(33,34);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(22,28)(23,24)(26,34)(27,35)(29,30)(31,36);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24,25,23)(22,27,29,31)(26,33,34,32)(28,36,30,35);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,24)(27,31)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7073182);