#857492

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2106 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=hi,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, 764688);