#9913678

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

4125 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)(3,5,11,7)(4,8,12,15)(10,13,16,14)(17,18,22,28)(19,26,30,23)(20,25,31,24)(21,29,32,27)(33,34,38,42)(35,39,44,48)(36,45,46,40)(37,41,43,47);(4,12)(8,15)(10,16)(13,14)(17,19)(18,23)(20,29)(21,25)(22,30)(24,32)(26,28)(27,31)(33,35,38,44)(34,39,42,48)(36,43,46,37)(40,41,45,47);(1,3,6,11)(2,7,9,5)(4,10,12,16)(8,14,15,13)(17,20,22,31)(18,24,28,25)(19,29,30,27)(21,23,32,26)(33,36,38,46)(34,40,42,45)(35,37,44,43)(39,47,48,41);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,23)(22,31)(24,28)(26,32)(27,30)(33,36)(34,40)(35,43)(37,44)(38,46)(39,41)(42,45)(47,48);(1,4,6,12)(2,8,9,15)(3,10,11,16)(5,13,7,14)(17,19)(18,26)(20,29)(21,24)(22,30)(23,28)(25,32)(27,31)(33,37,38,43)(34,41,42,47)(35,36,44,46)(39,45,48,40);(1,5)(2,3)(4,13)(6,7)(8,10)(9,11)(12,14)(15,16)(17,21,22,32)(18,27,28,29)(19,25,30,24)(20,23,31,26)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820874);