#484182

order	512
nilpotency class	4
isoclinism	isoclinic class of #484181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1170 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,27)(19,34)(20,32)(21,30)(23,33)(26,31)(28,36)(29,38)(35,39)(37,40);(2,5)(3,8)(6,7)(9,11,14,21)(10,15,18,30)(12,19,23,35)(13,20,27,38)(16,28,31,40)(17,32,25,29)(22,39,33,34)(24,36,26,37);(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,33)(18,31)(20,37)(21,35)(22,25)(24,27)(29,39)(30,40)(32,34)(36,38);(12,23)(16,31)(19,35)(22,33)(24,26)(28,40)(34,39)(36,37)

Polycyclic presentation:

2^9|b^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391378);