#9390564

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

4042 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297760);