#743543

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1740 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)(9,10,13,14)(11,15,16,12)(17,18,22,21)(19,24,20,23)(25,26,30,29)(27,32,28,31);(9,11,13,16)(10,12,14,15)(17,18,22,21)(19,23,20,24)(25,27,30,28)(26,31,29,32);(10,14)(11,16)(17,19,22,20)(18,23,21,24)(25,26,30,29)(27,31,28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20,22,19)(18,24,21,23)(25,28,30,27)(26,31,29,32);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,18)(19,24,20,23)(25,29,30,26)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=fg,d^2=fg,e^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 650739);