#9970161

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

4047 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)(19,23)(20,25)(21,26)(22,27)(24,29)(28,31)(30,32)(33,34,38,43)(35,41,46,39)(36,40,42,37)(44,45,48,47);(4,11)(8,14)(10,15)(12,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,27)(28,30)(31,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(2,7)(3,9)(8,14)(10,15)(17,19,22,29)(18,23,27,24)(20,28,30,26)(21,25,31,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19)(18,24)(20,28)(21,25)(22,29)(23,27)(26,30)(31,32)(33,36,38,42)(34,40,43,37)(35,44,46,48)(39,47,41,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,31)(26,29)(27,32)(33,35)(34,41)(36,44)(37,47)(38,46)(39,43)(40,45)(42,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,26)(19,25)(20,23)(22,31)(24,30)(27,28)(29,32)(33,37,38,40)(34,42,43,36)(35,45,46,47)(39,48,41,44)

Polycyclic presentation:

2^9|a^2=g,c^2=i,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, 9877357);