#7093341

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,29,33,47)(20,25,21,26)(22,38,30,28)(24,39,42,34)(31,46,35,41)(32,43,48,40)(36,45,37,44);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,30,39,23,33,22,34)(18,24,38,47,27,42,28,29)(20,35,37,48,21,31,36,32)(25,43,45,46,26,40,44,41);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,31)(21,23)(22,36)(24,40)(26,27)(28,44)(29,46)(30,37)(32,39)(33,35)(34,48)(38,45)(41,47)(42,43);(11,16)(13,15)(17,21,23,20)(18,26,27,25)(19,32,33,48)(22,36,30,37)(24,41,42,46)(28,44,38,45)(29,43,47,40)(31,34,35,39);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(18,27)(19,33)(20,21)(28,38)(34,39)(36,37)(40,43)(41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=ghi,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=df,e^b=eh,d^c=dh,e^c=ei,e^d=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, 7000537);