#844837

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
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

1848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fgi,d^2=h,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752033);