#9184716

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

4126 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)(3,8)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,31)(29,30)(33,34,38,42)(35,39,45,40)(36,37,41,47)(43,44,46,48);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19,22,21)(18,23,24,26)(20,28,30,27)(25,32,29,31)(33,35)(34,39)(36,43)(37,44)(38,45)(40,42)(41,46)(47,48);(2,4)(3,9)(10,12)(11,16)(18,24)(23,26)(25,29)(31,32)(33,35,38,45)(34,39,42,40)(36,43,41,46)(37,44,47,48);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,18)(19,23)(20,29)(21,26)(22,24)(25,30)(27,32)(28,31)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,20,22,30)(18,25,24,29)(19,27,21,28)(23,31,26,32)(33,36)(34,37)(35,43)(38,41)(39,44)(40,48)(42,47)(45,46);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,19)(18,26,24,23)(20,27,30,28)(25,31,29,32)(33,37)(34,41)(35,44)(36,42)(38,47)(39,46)(40,43)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=gh,e^2=g,f^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, 9091912);