#7093429

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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)(17,18,24,29)(19,30,21,27)(20,33,36,26)(22,28,23,39)(25,37,41,32)(31,42,34,45)(35,44,38,43)(40,47,48,46);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,37,24,21,23,32)(18,25,39,27,29,41,28,30)(20,31,38,46,36,34,35,47)(26,40,43,42,33,48,44,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,34)(22,38)(23,35)(24,36)(25,40)(27,42)(28,44)(29,33)(30,45)(32,47)(37,46)(39,43)(41,48);(11,16)(13,15)(17,21,24,19)(18,27,29,30)(20,34,36,31)(22,37,23,32)(25,28,41,39)(26,42,33,45)(35,47,38,46)(40,44,48,43);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,29,39)(19,32,21,37)(20,35,36,38)(25,27,41,30)(26,43,33,44)(31,46,34,47)(40,45,48,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,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=ehi,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, 7000625);