#382821

order	512
nilpotency class	3
isoclinism	isoclinic class of #213788
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1203 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)(11,16)(12,17)(13,29)(14,18)(15,19)(20,30)(21,35)(22,31)(23,32)(24,39)(25,26)(27,40)(28,34)(33,38)(36,37);(9,11,15,23)(10,16,19,32)(12,24,26,38)(13,22,14,21)(17,33,25,39)(18,35,29,31)(20,28,37,27)(30,40,36,34);(2,5)(6,8)(9,12,14,28,15,26,13,27)(10,17,18,34,19,25,29,40)(11,20,22,38,23,37,21,24)(16,30,31,33,32,36,35,39);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,25)(13,18)(14,29)(15,19)(17,26)(20,36)(21,31)(22,35)(23,32)(24,39)(27,40)(28,34)(30,37)(33,38)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,h^b=hi,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, 290017);