#376240

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6,8,7)(9,10)(11,21)(12,24)(13,16)(14,29)(15,20)(17,27)(18,23)(19,30)(22,33)(25,37)(26,34)(28,32)(31,39)(35,36)(38,40);(1,2)(3,4)(9,11,14,13,15,23,30,28)(10,16,19,21,20,32,29,18)(12,22,26,25,27,36,40,39)(17,31,34,35,24,37,38,33);(2,4)(9,12,14,26,15,27,30,40)(10,17,19,34,20,24,29,38)(11,22,13,25,23,36,28,39)(16,31,21,35,32,37,18,33);(5,7,8,6)(9,13,14,23,15,28,30,11)(10,18,19,16,20,21,29,32)(12,25,26,36,27,39,40,22)(17,33,34,31,24,35,38,37)

Polycyclic presentation:

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

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, 283436);