#9970164

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

4362 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(4,11)(8,14)(10,15)(12,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(34,38)(39,45)(40,43)(46,47);(2,7)(3,9)(8,14)(10,15)(18,20)(19,26)(25,28)(27,32)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,35)(34,39)(36,41)(37,42)(38,45)(40,46)(43,47)(44,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,36,37,44)(34,40,38,43)(35,41,42,48)(39,46,45,47)

Polycyclic presentation:

2^9|c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,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, 9877360);