#1301615

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

1708 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fg,d^2=fgi,e^2=fgh,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, 1208811);