#678217

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

1636 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=f,d^2=gi,e^2=fg,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, 585413);