#1287183

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1715 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgh,c^2=ghi,d^2=f,e^2=fg,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, 1194379);