#7089948

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	#24265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(19,27,30,42)(20,31,33,24)(21,35,37,26)(23,36,39,29)(28,44,32,43)(34,41,48,47)(38,46,40,45);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,25,39)(20,32,33,28)(21,36,37,29)(24,40,31,38)(26,42,35,27)(34,46,48,45)(41,43,47,44);(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)(21,34,37,48)(23,38,39,40)(26,41,35,47)(27,43,42,44)(29,46,36,45);(11,16)(13,15)(19,29)(20,33)(21,37)(23,27)(24,31)(26,35)(28,45)(30,36)(32,46)(38,44)(39,42)(40,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,25)(19,30)(20,33)(26,35)(29,36)(34,48)(38,40)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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=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, 6997144);