#2470448

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=g,d^2=g,e^2=hi,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, 2377644);