#383642

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290838);