#865717

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
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

1858 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 772913);