#6489085

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

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,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, 6396281);