#6474119

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,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, 6381315);