#382460

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2236 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289656);