#4632824

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
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

1382 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fgi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4540020);