#371328

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1196 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,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, 278524);