#392260

order	512
nilpotency class	3
isoclinism	isoclinic class of #184979
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3378 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)(9,10,14,20,15,18,13,19)(11,21,23,34,24,16,29,31)(12,25,27,17,28,33,26,32)(22,36,37,30,38,40,39,35);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,21)(19,31)(20,34)(25,40)(26,37)(27,39)(28,38)(32,35)(33,36);(9,12,15,28)(10,17,18,32)(11,22,24,38)(13,27,14,26)(16,30,21,35)(19,33,20,25)(23,39,29,37)(31,36,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299456);