#7090612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 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)(17,18,22,25)(19,27,30,43)(20,31,33,24)(21,35,36,26)(23,37,39,29)(28,44,32,42)(34,41,45,48)(38,47,40,46);(9,10)(11,15)(12,14)(13,16)(17,19,21,29,22,30,36,37)(18,23,26,27,25,39,35,43)(20,32,34,46,33,28,45,47)(24,40,41,44,31,38,48,42);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,22,33)(18,24,25,31)(19,28,30,32)(21,34,36,45)(23,38,39,40)(26,41,35,48)(27,42,43,44)(29,47,37,46);(11,16)(13,15)(19,29)(20,33)(21,36)(23,27)(24,31)(26,35)(28,46)(30,37)(32,47)(38,44)(39,43)(40,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,25)(20,33)(23,39)(26,35)(27,43)(28,32)(34,45)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997808);