#537057

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

1970 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)(24,32)(25,26)(27,30)(28,34)(31,35)(33,36);(6,9)(10,12)(13,14,16,18)(15,17,19,20)(22,25)(23,30)(28,32)(31,33);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,33)(25,29)(26,30)(28,36)(31,34)(32,35);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,17,18,20)(21,24)(22,28)(23,31)(25,32)(26,34)(27,35)(29,36)(30,33);(21,25,26,22)(23,29,30,27)(24,32,34,28)(31,36,33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fgi,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=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, 444253);