#581482

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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

1894 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)(5,6)(7,8)(9,10)(11,12)(14,17)(15,19)(21,22,26,28)(23,29,30,24)(25,33,34,27)(31,32,36,35);(1,2,4,3)(6,10)(7,12)(13,14)(15,20)(16,17)(18,19)(23,30)(24,29)(31,36)(32,35);(6,10)(8,11)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,31)(26,30)(27,32)(28,29)(33,35)(34,36);(1,3,4,2)(5,7,9,12)(6,11,10,8)(13,15)(14,18)(16,19)(17,20)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,8,9,11)(6,12,10,7)(13,14)(15,18)(16,17)(19,20)(21,25,26,34)(22,27,28,33)(23,31,30,36)(24,32,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fgh,d^2=fhi,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 488678);