#7188520

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1226 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)(4,8)(5,7)(10,14)(12,15)(17,18,22,26)(19,27,30,38)(20,24,33,41)(21,35,36,25)(23,29,39,31)(28,43,32,46)(34,48,47,42)(37,45,40,44);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,12,16,15)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,38)(26,39)(27,35)(28,33)(31,36)(34,44)(37,41)(42,43)(45,47)(46,48);(2,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,24)(19,28)(21,34)(22,33)(23,37)(25,42)(26,41)(27,43)(29,45)(30,32)(31,44)(35,48)(36,47)(38,46)(39,40);(1,4,5,3)(2,8,7,6)(10,14)(11,16)(19,29)(20,33)(21,36)(23,38)(24,41)(25,35)(27,39)(28,44)(30,31)(32,45)(37,43)(40,46);(9,12,13,15)(10,11,14,16)(19,30)(20,33)(23,39)(24,41)(27,38)(29,31)(34,47)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7095716);