#480639

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

848 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=chi,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, 387835);