#521886

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1962 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=fg,e^2=fgh,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, 429082);