#9039259

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

4310 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)(3,8)(4,9)(5,10)(6,7)(11,12)(13,15)(14,16)(19,25)(23,26)(27,31)(30,32)(33,34)(35,40)(36,39)(37,41)(38,46)(42,48)(43,44)(45,47);(2,7)(4,10)(8,15)(11,14)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35)(34,40)(36,43)(37,38)(39,44)(41,46)(42,45)(47,48);(4,10)(5,9)(11,14)(12,16)(18,22)(20,28)(23,26)(27,31)(33,34,39,36)(35,43,44,40)(37,45,47,41)(38,46,48,42);(1,3)(2,8)(4,14)(5,12)(6,13)(7,15)(9,16)(10,11)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,39,34)(35,43,44,40)(37,41,47,45)(38,46,48,42);(1,4,6,10)(2,9,7,5)(3,11,13,14)(8,12,15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,37)(34,41)(35,38)(36,45)(39,47)(40,46)(42,43)(44,48);(1,5,6,9)(2,10,7,4)(3,12,13,16)(8,11,15,14)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,38)(34,42)(35,37)(36,46)(39,48)(40,45)(41,43)(44,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8946455);