#620158

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

1934 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)(3,7)(4,6)(5,8)(10,13)(11,15)(17,18,20,22)(19,23,24,21)(25,26,29,32)(27,30,28,31);(1,2,4,6)(3,5,8,7)(9,10,14,13)(11,16,15,12)(25,27,29,28)(26,30,32,31);(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,31)(29,32);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,15,13,11)(17,19,20,24)(18,21,22,23)(25,28,29,27)(26,31,32,30);(1,3,4,8)(2,5,6,7)(9,13,14,10)(11,16,15,12)

Polycyclic presentation:

2^9|a^2=gh,b^2=fghi,d^2=g,e^2=f,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, 527354);