#893342

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=f,e^2=i,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, 800538);