#487376

order	512
nilpotency class	4
isoclinism	isoclinic class of #486875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,35)(28,37)(29,36)(30,39)(31,41)(32,43)(33,40)(34,38)(42,46)(44,47)(45,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,19,22,30)(18,23,27,39)(20,31,32,28)(21,34,35,29)(24,36,26,38)(25,41,43,37)(33,44,46,45)(40,48,42,47);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(18,24)(20,32)(21,35)(23,36)(25,42)(26,27)(28,31)(29,34)(37,48)(38,39)(40,43)(41,47);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(17,20,22,32)(18,25,27,43)(19,28,30,31)(21,33,35,46)(23,37,39,41)(24,40,26,42)(29,44,34,45)(36,47,38,48)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394572);