#379963

order	512
nilpotency class	3
isoclinism	isoclinic class of #196257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287159);