#1499559

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1406755);