#678059

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=g,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 585255);