#856512

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1660 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=fgi,d^2=f,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, 763708);