#7083447

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 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)(17,18,22,25)(19,27,32,37)(20,24,33,40)(21,35,28,26)(23,30,38,31)(29,41,44,47)(34,46,42,39)(36,43,48,45);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,28,30,22,32,21,31)(18,23,35,37,25,38,26,27)(20,29,42,43,33,44,34,45)(24,36,46,47,40,48,39,41);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,29)(21,34)(22,33)(23,36)(25,40)(26,39)(27,41)(28,42)(30,43)(31,45)(32,44)(35,46)(37,47)(38,48);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(19,30)(21,28)(23,37)(26,35)(27,38)(29,43)(31,32)(34,42)(36,47)(39,46)(41,48)(44,45);(10,14)(13,15)(18,25)(20,33)(23,38)(26,35)(27,37)(29,44)(34,42)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=gh,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 6990643);