#363724

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1091 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270920);