#159802

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

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, 66998);