#6493790

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1515 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)(3,5)(4,6)(7,8)(11,15)(14,16)(17,18,22,27,23,28,34,41)(19,29,32,40,33,44,37,24)(20,25,36,43,35,42,21,26)(30,39,31,45,46,48,47,38);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,31)(22,37)(23,33)(25,38)(26,39)(27,44)(28,40)(29,41)(32,34)(35,46)(36,47)(42,45)(43,48);(2,6)(5,8)(10,13)(14,16)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,38)(26,41)(27,43)(28,42)(29,39)(31,32)(33,46)(37,47)(40,45)(44,48);(10,13)(14,16)(17,21,23,36)(18,26,28,43)(19,31,33,47)(20,34,35,22)(24,39,40,48)(25,41,42,27)(29,45,44,38)(30,32,46,37);(9,11,12,15)(10,14,13,16)(20,35)(21,36)(25,42)(26,43)(30,46)(31,47)(38,45)(39,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,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, 6400986);