#9166773

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

4517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,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, 9073969);