#484532

order	512
nilpotency class	4
isoclinism	isoclinic class of #484181
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
prime radical	#537	C₂³
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

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^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,c^b=ci,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, 391728);