#6718108

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18,22,23)(19,24,30,29)(20,31,34,25)(21,26,36,37)(27,32,43,38)(28,39,35,41)(33,44,47,40)(42,45,48,46);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,24,23,29)(20,32,34,38)(21,35,36,28)(25,27,31,43)(26,41,37,39)(33,46,47,45)(40,48,44,42);(10,15)(14,16)(17,20,22,34)(18,25,23,31)(19,27,30,43)(21,33,36,47)(24,38,29,32)(26,40,37,44)(28,42,35,48)(39,46,41,45);(2,6)(5,7)(17,21)(18,26)(19,28)(20,33)(22,36)(23,37)(24,39)(25,40)(27,42)(29,41)(30,35)(31,44)(32,45)(34,47)(38,46)(43,48);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(18,23)(19,29)(21,36)(24,30)(25,31)(27,32)(28,39)(33,47)(35,41)(38,43)(42,46)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6625304);