#385283

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ef,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292479);