#631761

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2079 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=gi,d^2=gi,e^2=gh,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, 538957);