#522243

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 429439);