#6472907

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,33)(21,26)(22,38)(23,28)(24,31)(25,30)(27,39)(32,44)(34,42)(35,41)(36,40)(37,46)(43,47)(45,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,32,39,34)(18,24,27,29,28,42,38,44)(20,21,30,37,35,36,47,48)(25,40,33,46,43,26,41,45);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,30,23,35,39,47)(18,25,27,33,28,43,38,41)(19,21,31,37,32,36,34,48)(24,40,29,46,42,26,44,45);(11,15)(13,16)(17,21)(18,26)(19,30)(20,34)(22,37)(23,36)(24,41)(25,29)(27,45)(28,40)(31,35)(32,47)(33,42)(38,46)(39,48)(43,44);(10,14)(13,16)(20,35)(21,36)(25,43)(26,40)(30,47)(33,41)(37,48)(45,46)

Polycyclic presentation:

2^9|b^2=g,c^2=f,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,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6380103);