#9039447

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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gi,e^2=gh,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, 8946643);