#9947419

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

4135 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)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(2,7)(3,9)(10,13)(11,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(35,41)(38,45)(42,47)(46,48);(2,7)(3,9)(10,13)(11,16)(18,25)(19,29)(20,31)(27,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gi,e^2=h,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, 9854615);