#6476131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1676 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,33,44)(20,25,35,42)(21,26,36,43)(22,37,34,27)(24,38,41,32)(30,45,46,40)(31,39,47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,38)(23,33)(25,39)(26,40)(27,44)(28,41)(29,37)(32,34)(35,46)(36,47)(42,48)(43,45);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,39)(26,27)(28,42)(29,45)(31,32)(33,46)(37,43)(38,47)(40,44)(41,48);(1,4,5,3)(2,7,8,6)(11,15)(13,16)(17,21,23,36)(18,26,28,43)(19,31,33,47)(20,34,35,22)(24,40,41,45)(25,27,42,37)(29,39,44,48)(30,32,46,38);(10,14)(13,16)(20,35)(21,36)(25,42)(26,43)(30,46)(31,47)(39,48)(40,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6383327);