#576806

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

1758 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)(5,6,9,10)(7,8,12,11)(21,22)(23,27)(24,30)(25,33)(26,29)(28,34)(31,36)(32,35);(6,10)(7,12)(14,17)(18,20)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(6,10)(8,11)(13,14)(15,18)(16,17)(19,20)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(5,7)(6,11)(8,10)(9,12)(13,15,16,19)(14,18,17,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(5,8,9,11)(6,7,10,12)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=h,e^2=f,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, 484002);