#6744185

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1283 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)(9,10,12,13)(11,15,16,14)(17,18)(19,27)(20,32)(21,25)(22,26)(23,31)(24,33)(28,38)(29,37)(30,36)(34,45)(35,40)(39,46)(41,44)(42,47)(43,48);(10,13)(11,16)(17,19)(18,23)(20,29)(21,31)(22,28)(24,37)(25,27)(26,36)(30,35)(32,41)(33,44)(34,42)(38,40)(39,47)(43,46)(45,48);(2,6)(7,8)(9,11,12,16)(10,14,13,15)(19,28)(20,33)(22,35)(23,36)(24,32)(26,40)(27,38)(29,43)(30,31)(37,48)(41,47)(42,44);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,21,33)(18,24,25,32)(19,29,31,44)(22,34,35,46)(23,37,27,41)(26,39,40,45)(28,42,30,43)(36,47,38,48);(1,4,5,3)(2,8,6,7)(18,25)(19,30)(22,35)(23,36)(24,32)(27,38)(28,31)(29,43)(34,46)(37,47)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6651381);