#1313128

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1554 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)(3,8)(4,7)(5,6)(11,16)(12,14)(17,18,22,20)(19,24,23,21);(1,3,6,4)(2,7,5,8)(9,10,13,15)(11,14,16,12)(26,29)(30,32);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,4,6,3)(2,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,18)(19,24,23,21)(25,27,28,31)(26,30,29,32);(1,5,6,2)(3,8,4,7)(9,12,13,14)(10,11,15,16)(17,21,22,24)(18,23,20,19)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1220324);