#641130

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1826 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 548326);