#359311

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efi,c^2=i,d^2=eh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266507);