#7089945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1441 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,28)(19,29,33,42)(20,34,21,26)(22,27,23,38)(25,39,43,32)(30,46,36,41)(31,40,47,48)(35,45,37,44);(9,10)(11,15)(12,14)(13,16)(17,19,24,33)(18,25,28,43)(20,30,21,36)(22,32,23,39)(26,40,34,48)(27,42,38,29)(31,37,47,35)(41,45,46,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,21)(18,26,28,34)(19,30,33,36)(22,37,23,35)(25,40,43,48)(27,45,38,44)(29,41,42,46)(31,32,47,39);(11,16)(13,15)(17,21,24,20)(18,26,28,34)(19,31,33,47)(22,37,23,35)(25,41,43,46)(27,44,38,45)(29,40,42,48)(30,32,36,39);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,27,28,38)(19,32,33,39)(20,35,21,37)(25,42,43,29)(26,44,34,45)(30,31,36,47)(40,41,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997141);