#4928745

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

1509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=ghi,c^2=ghi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=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, 4835941);