#517896

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fgh,d^2=i,e^2=fg,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, 425092);