#9174999

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9082195);