#6494986

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(4,5)(6,8)(11,15)(14,16)(17,18,23,37,22,27,21,28)(19,29,36,24,32,44,31,41)(20,26,38,42,34,25,35,43)(30,46,48,40,33,45,47,39);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,32)(18,24,27,41)(20,33,34,30)(21,36,23,31)(25,40,26,39)(28,44,37,29)(35,47,38,48)(42,45,43,46);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,38)(24,39)(26,27)(28,42)(29,45)(31,47)(32,33)(36,48)(37,43)(40,41)(44,46);(10,13)(14,16)(17,20)(18,26)(19,30)(21,35)(22,34)(23,38)(24,40)(25,27)(28,43)(29,46)(31,47)(32,33)(36,48)(37,42)(39,41)(44,45);(9,11,12,15)(10,14,13,16)(18,27)(20,34)(24,41)(28,37)(29,44)(30,33)(35,38)(47,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gi,e^2=hi,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, 6402182);