#9481385

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

3929 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388581);