#6477764

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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(18,23)(19,27)(21,36)(25,38)(26,39)(29,43)(30,31)(32,42)(33,34)(37,41)(45,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18,21,23,22,26,36,39)(19,28,30,41,31,24,27,37)(20,32,34,42,35,25,33,38)(29,40,45,47,46,44,43,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,19,22,31)(18,24,26,28)(20,29,35,46)(21,30,36,27)(23,37,39,41)(25,40,32,44)(33,43,34,45)(38,47,42,48);(3,10)(6,14)(9,11)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,38)(24,40)(26,32)(27,43)(28,44)(30,45)(31,46)(33,36)(37,47)(39,42)(41,48);(2,8)(6,14)(7,16)(13,15)(17,21,22,36)(18,23,26,39)(19,30,31,27)(20,33,35,34)(24,37,28,41)(25,42,32,38)(29,43,46,45)(40,48,44,47)

Polycyclic presentation:

2^9|b^2=f,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, 6384960);