#2152738

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

1486 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)(10,12)(11,15)(25,26,30,31)(27,28,32,35)(29,33,38,39)(34,36,37,40);(1,3,6,4)(2,7,5,8)(9,10,13,12)(11,16,15,14)(17,18,21,20)(19,23,24,22)(26,31)(28,35)(33,39)(36,40);(2,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19,21,24)(18,22,20,23)(25,27,30,32)(26,28,31,35)(29,37,38,34)(33,40,39,36);(1,4,6,3)(2,8,5,7)(17,20,21,18)(19,23,24,22)(25,28)(26,32)(27,31)(29,36)(30,35)(33,37)(34,39)(38,40);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,14,15,16)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=fgh,d^2=gh,e^2=g,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, 2059934);