#1301614

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

1574 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,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, 1208810);