#9783043

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
prime radical	#50847	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,e^2=g,f^2=g,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, 9690239);