#9720506

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

4131 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9627702);