#6484615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473167
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×#3179	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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)(17,18,23,21)(19,27,29,24)(20,31,34,46)(22,36,37,26)(25,40,35,33)(28,43,32,47)(30,39,45,42)(38,44,41,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,22,30,23,29,37,45)(18,24,26,39,21,27,36,42)(20,32,33,48,34,28,40,44)(25,41,31,43,35,38,46,47);(3,9)(6,13)(10,11)(14,15)(17,20,22,33,23,34,37,40)(18,25,26,31,21,35,36,46)(19,28,30,44,29,32,45,48)(24,38,39,47,27,41,42,43);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21,23,18)(19,24,29,27)(20,25,34,35)(22,36,37,26)(28,41,32,38)(30,39,45,42)(31,40,46,33)(43,48,47,44);(2,8)(6,13)(7,16)(14,15)(18,21)(24,27)(25,35)(26,36)(31,46)(38,41)(39,42)(43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6391811);