#6477439

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei

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, 6384635);