#645663

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1926 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fg,d^2=fh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df

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, 552859);