#517724

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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

1990 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424920);