#484857

order	512
nilpotency class	4
isoclinism	isoclinic class of #484601
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

795 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,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,e^c=eh,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, 392053);