#9053874

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

4481 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,e^2=gi,f^2=g,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, 8961070);