#6477765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3199	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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)(18,24)(19,29)(21,22)(26,39)(27,28)(31,43)(32,33)(34,42)(35,37)(38,41)(45,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18,21,27,23,28,22,24)(19,30,29,38,33,25,32,41)(20,34,37,39,36,26,35,42)(31,40,43,48,46,44,45,47);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,19,23,33)(18,25,28,30)(20,31,36,46)(21,29,22,32)(24,38,27,41)(26,40,34,44)(35,45,37,43)(39,47,42,48);(3,10)(6,14)(9,11)(13,15)(17,20)(18,26)(19,31)(21,37)(22,35)(23,36)(24,39)(25,40)(27,42)(28,34)(29,43)(30,44)(32,45)(33,46)(38,47)(41,48);(2,8)(6,14)(7,16)(13,15)(17,21,23,22)(18,27,28,24)(19,29,33,32)(20,35,36,37)(25,41,30,38)(26,39,34,42)(31,45,46,43)(40,47,44,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,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,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, 6384961);