#9054411

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

4827 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961607);