#9939580

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

4182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9846776);