#382827

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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)(11,17)(12,24)(13,19)(14,20)(15,30)(16,21)(18,26)(22,38)(23,34)(25,35)(27,36)(28,32)(29,33)(31,37)(39,40);(9,11,14,13)(10,17,20,19)(12,25,26,39)(15,29,16,23)(18,35,24,40)(21,34,30,33)(22,31,28,27)(32,37,38,36);(2,5)(6,8)(9,12,16,27,14,26,15,31)(10,18,21,37,20,24,30,36)(11,22,23,39,13,28,29,25)(17,32,34,35,19,38,33,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,19,20,17)(12,22,26,28)(15,29,16,23)(18,32,24,38)(21,34,30,33)(25,31,39,27)(35,37,40,36)

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,d^c=de,e^c=ei,g^c=gi,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, 290023);