#877569

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2130 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(18,22)(20,23)(25,26,31,28)(27,29,32,30);(2,6)(7,8)(9,10,12,14)(11,13,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,23,22,20)(25,28,31,26)(27,29,32,30);(1,2)(3,7)(4,8)(5,6)(25,29,31,30)(26,27,28,32);(1,4,5,3)(2,8,6,7)(9,11,12,16)(10,13,14,15)(17,20,21,23)(18,24,22,19)(25,30,31,29)(26,32,28,27)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fh,d^2=i,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, 784765);