#9039397

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

3975 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gi,e^2=h,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, 8946593);