#6489084

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1687 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)(11,14)(12,15)(13,16)(17,18)(19,30)(20,35)(21,26)(22,28)(23,27)(24,29)(25,33)(31,46)(32,45)(34,44)(36,42)(37,43)(38,40)(39,47)(41,48);(11,12)(14,15)(17,19,24,34)(18,25,29,40)(20,32,21,31)(22,33,23,38)(26,41,35,39)(27,30,28,44)(36,48,37,47)(42,45,43,46);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20,24,21)(18,26,29,35)(19,31,34,32)(22,37,23,36)(25,39,40,41)(27,43,28,42)(30,45,44,46)(33,48,38,47);(9,12,13,11)(10,15,16,14)(17,21,24,20)(18,26,29,35)(19,32,34,31)(22,36,23,37)(25,39,40,41)(27,43,28,42)(30,45,44,46)(33,47,38,48);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,22,24,23)(18,27,29,28)(19,33,34,38)(20,36,21,37)(25,30,40,44)(26,42,35,43)(31,47,32,48)(39,46,41,45)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=h,e^2=g,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, 6396280);