#790120

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

1756 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,24)(19,20,23,22)(25,26,29,28)(27,31,32,30);(2,6)(4,7)(9,11,13,16)(10,14,12,15)(25,26,29,28)(27,30,32,31);(9,10,13,12)(11,15,16,14)(17,19,21,23)(18,22,24,20)(25,27,29,32)(26,30,28,31);(1,3,5,8)(2,7,6,4)(9,12,13,10)(11,15,16,14)(17,18)(19,22)(20,23)(21,24);(1,4,5,7)(2,3,6,8)(9,11)(10,15)(12,14)(13,16)(17,20,21,22)(18,23,24,19)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=g,d^2=f,e^2=fg,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, 697316);