#7083545

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1495 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,21)(19,27,30,40)(20,25,32,31)(22,35,37,26)(24,36,34,29)(28,41,42,48)(33,45,46,39)(38,47,44,43);(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,24,21,34)(20,28,32,42)(22,36,37,29)(25,38,31,44)(26,40,35,27)(33,47,46,43)(39,48,45,41);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,31)(22,33)(23,32)(24,38)(26,39)(27,41)(29,43)(30,42)(34,44)(35,45)(36,47)(37,46)(40,48);(11,16)(13,15)(17,21,23,18)(19,27,30,40)(20,31,32,25)(22,26,37,35)(24,36,34,29)(28,41,42,48)(33,39,46,45)(38,47,44,43);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(20,32)(25,31)(28,42)(33,46)(38,44)(39,45)(41,48)(43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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,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, 6990741);