#1286078

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

1674 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=i,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, 1193274);