#383473

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 290669);