#4509935

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1524 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4417131);