#640965

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

1677 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)(3,5)(4,7)(6,8)(17,18,21,20)(19,23,24,22)(27,31)(29,32);(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,13,15,16)(17,18,21,20)(19,23,24,22);(2,3)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,26,28,30)(27,29,31,32);(1,3,5,2)(4,7,8,6)(17,18,21,20)(19,22,24,23)(25,27)(26,29)(28,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,23,24,22)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=gi,d^2=g,e^2=fh,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, 548161);