#741873

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

1860 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=gi,d^2=fghi,e^2=fgh,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, 649069);