#486096

order	512
nilpotency class	4
isoclinism	isoclinic class of #486091
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×C₂²⋊C₄	C₂⁴
prime radical	#1595	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

807 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,24)(20,25)(21,23)(22,26)(27,38)(28,35)(29,39)(30,40)(31,36)(32,41)(33,45)(34,37)(42,46)(43,47)(44,48);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(18,23)(24,35)(25,36)(26,37)(38,46)(39,45)(40,47)(41,48);(17,19,22,29)(18,24,26,39)(20,30,32,27)(21,33,34,28)(23,35,37,45)(25,40,41,38)(31,42,44,43)(36,47,48,46);(2,7)(3,10)(8,12)(11,13)(17,20,22,32)(18,25,26,41)(19,27,29,30)(21,31,34,44)(23,36,37,48)(24,38,39,40)(28,42,33,43)(35,46,45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393292);