#6381421

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi

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, 6288617);