#6477763

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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,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, 6384959);