#2227995

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,8)(4,7)(5,6)(11,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(10,13)(14,16)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(2,5)(3,7)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,4,6,8)(2,7,5,3)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,5,6,2)(3,8,7,4)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=fgi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2135191);