#6731228

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1265 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,27)(19,24,31,38)(20,25,33,42)(21,34,22,26)(28,36,30,39)(29,37,44,48)(32,41,35,40)(43,46,45,47);(10,13)(11,15)(17,19,23,31)(18,24,27,38)(20,29,33,44)(21,30,22,28)(25,37,42,48)(26,36,34,39)(32,43,35,45)(40,47,41,46);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(19,28)(21,22)(24,36)(26,34)(29,43)(30,31)(32,35)(37,46)(38,39)(40,41)(44,45)(47,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,35)(22,32)(23,33)(24,37)(26,41)(27,42)(28,43)(30,45)(31,44)(34,40)(36,46)(38,48)(39,47);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(20,32)(22,23)(24,38)(25,40)(27,34)(28,30)(29,44)(33,35)(41,42)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

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, 6638424);