#630309

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

1699 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=fhi,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 537505);