#9794352

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

4223 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)(19,26)(20,29)(21,24)(22,30)(23,25)(27,28)(31,32)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,21)(18,24,25,26)(20,27,30,31)(22,28,29,32)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(2,6)(7,9)(8,12)(14,15)(18,25)(20,30)(24,26)(27,31)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,22)(19,27)(21,31)(23,30)(24,28)(25,29)(26,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,19)(18,24,25,26)(20,31,30,27)(22,28,29,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,30)(26,31)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9701548);