#9947264

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

4133 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,34)(35,44)(36,37)(38,48)(39,41)(40,45)(42,47)(43,46);(2,7)(3,9)(10,13)(11,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,35,39,45)(34,40,41,44)(36,46,47,38)(37,48,42,43);(2,7)(3,9)(10,13)(11,16)(34,41)(35,45)(36,47)(43,48);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,36)(34,42)(35,46)(37,41)(38,45)(39,47)(40,43)(44,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30)(33,37,39,42)(34,36,41,47)(35,43,45,48)(38,44,46,40);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)(33,38,39,46)(34,43,41,48)(35,36,45,47)(37,44,42,40)

Polycyclic presentation:

2^9|b^2=ghi,d^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, 9854460);