#480493

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

871 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,f^b=fg,d^c=di,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387689);