#9794339

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

4235 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,28)(20,30)(21,25)(22,29)(23,24)(26,31)(27,32)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(2,6)(7,9)(8,12)(14,15)(18,24)(19,21)(26,30)(27,29)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(17,19,23,21)(18,25,24,28)(20,27,31,29)(22,26,32,30)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,20,23,31)(18,26,24,30)(19,29,21,27)(22,28,32,25)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,21,23,19)(18,25,24,28)(20,27,31,29)(22,30,32,26)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22)(18,27)(19,26)(20,28)(21,30)(23,32)(24,29)(25,31)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,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, 9701535);