#2208117

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1628 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=fgh,e^2=fg,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, 2115313);