#857824

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

1882 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=h,e^2=f,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, 765020);