#9175040

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

4292 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,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, 9082236);