#6476946

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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,23,27,37,42)(19,28,31,39,32,41,36,24)(20,33,34,48,21,25,35,40)(29,44,45,38,30,43,46,47);(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,24,27,39)(20,29,21,30)(22,36,37,31)(25,38,33,47)(26,41,42,28)(34,46,35,45)(40,44,48,43);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20,23,21)(18,25,27,33)(19,29,32,30)(22,34,37,35)(24,38,39,47)(26,40,42,48)(28,43,41,44)(31,45,36,46);(11,15)(13,16)(17,21,23,20)(18,25,27,33)(19,30,32,29)(22,35,37,34)(24,38,39,47)(26,40,42,48)(28,43,41,44)(31,46,36,45);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(20,21)(25,33)(29,30)(34,35)(38,47)(40,48)(43,44)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6384142);