#9910222

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

4115 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)(3,6)(4,11)(5,8)(7,12)(9,15)(10,14)(13,16)(17,18)(19,27)(20,25)(21,24)(22,31)(23,26)(28,29)(30,32)(35,44)(37,47)(39,40)(42,45);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19,23,21)(18,24,26,27)(20,28,31,30)(22,29,25,32)(33,34,38,43)(35,40,44,39)(36,41,46,48)(37,42,47,45);(1,2,5,8)(3,6,10,14)(4,7,12,11)(9,13,16,15)(17,20,23,31)(18,22,26,25)(19,28,21,30)(24,29,27,32)(33,35,38,44)(34,39,43,40)(36,45,46,42)(37,41,47,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,25)(19,28)(21,30)(22,26)(23,31)(24,32)(27,29)(33,35)(34,40)(36,45)(37,41)(38,44)(39,43)(42,46)(47,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21,23,19)(18,24,26,27)(20,30,31,28)(22,29,25,32)(33,36)(34,41)(35,45)(37,40)(38,46)(39,47)(42,44)(43,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,22,23,25)(18,20,26,31)(19,29,21,32)(24,28,27,30)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,45)(44,48)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=g,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, 9817418);