#382418

order	512
nilpotency class	3
isoclinism	isoclinic class of #184985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3558 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)(12,24)(13,18)(14,29)(15,19)(17,27)(20,35)(21,31)(22,38)(23,32)(25,39)(26,34)(28,40)(30,37)(33,36);(2,5)(6,8)(9,11)(10,16)(12,25)(13,21)(14,22)(15,23)(17,33)(18,31)(19,32)(20,28)(24,39)(26,37)(27,36)(29,38)(30,34)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17,19,34)(11,20,23,25)(13,26)(14,27)(15,28)(16,30)(18,24,29,40)(21,36,22,37)(31,35)(32,33)(38,39);(3,7)(6,8)(9,13,15,14)(10,18,19,29)(11,21,23,22)(12,26,28,27)(16,31,32,38)(17,24,34,40)(20,36,25,37)(30,35,33,39)

Polycyclic presentation:

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

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, 289614);