#850902

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

2114 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=g,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, 758098);