#391090

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=eh,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

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, 298286);