#518723

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

1998 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)(10,12)(13,14,16,18)(15,17,19,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,34)(33,35);(13,14,16,18)(15,17,19,20)(21,23,27,25)(22,28,24,30)(26,31,34,33)(29,35,32,36);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24,27,22)(23,28,25,30)(26,32,34,29)(31,35,33,36);(1,3,4,2)(21,25,27,23)(22,28,24,30)(26,31,34,33)(29,36,32,35);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,17,18,20)(21,26,27,34)(22,29,24,32)(23,31,25,33)(28,35,30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=hi,e^2=g,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, 425919);