#6476853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469989
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

1731 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,25,34,39)(19,27,29,36,30,40,33,23)(20,24,32,38,31,37,44,47)(28,41,43,46,42,48,45,35);(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,25,36)(20,28,31,42)(21,33,34,29)(24,35,37,46)(26,40,39,27)(32,45,44,43)(38,48,47,41);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20,22,31)(18,24,25,37)(19,28,30,42)(21,32,34,44)(23,35,36,46)(26,38,39,47)(27,41,40,48)(29,43,33,45);(11,15)(13,16)(18,25)(23,36)(24,37)(26,39)(27,40)(35,46)(38,47)(41,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(20,31)(24,37)(28,42)(32,44)(35,46)(38,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6384049);