#4368752

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

1586 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)(11,12)(15,16)(17,18,20,22)(19,23,24,21)(25,26,30,31)(27,28,32,36)(29,37,38,33)(34,40,39,35);(9,10,14,13)(11,15,12,16)(19,24)(21,23)(26,31)(27,32)(29,38)(35,40);(9,11,14,12)(10,15,13,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,16,13,15)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,37)(38,40);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,16,12,15)(17,18)(19,21)(20,22)(23,24)(25,29,30,38)(26,33,31,37)(27,34,32,39)(28,35,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=fgh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4275948);