#382814

order	512
nilpotency class	3
isoclinism	isoclinic class of #213736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1240 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)(11,15)(12,16)(13,18)(14,17)(19,25)(20,27)(21,26)(22,29)(23,30)(24,31)(28,32)(33,36)(34,37)(35,39)(38,40);(9,11,14,21)(10,15,17,26)(12,19,22,32)(13,20,24,34)(16,28,29,25)(18,27,31,37)(23,33,35,40)(30,38,39,36);(2,5)(6,8)(9,12,13,23,14,22,24,35)(10,16,18,30,17,29,31,39)(11,19,20,33,21,32,34,40)(15,25,27,36,26,28,37,38);(1,3,4,7)(2,6,5,8)(12,22)(13,24)(16,29)(18,31)(19,32)(20,34)(25,28)(27,37)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=f,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^d=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, 290010);