#892318

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

1722 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,d^2=gh,e^2=fghi,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, 799514);