#9980343

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

3946 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gh,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=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, 9887539);