#9058769

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
prime radical	#43433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,10,11,8)(4,9,14,5)(12,16,15,13)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,32)(28,31)(33,34,38,39)(35,41,43,36)(37,45,46,40)(42,44,48,47);(2,7)(3,11)(4,14)(13,16)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,35,38,43)(34,36,39,41)(37,42,46,48)(40,44,45,47);(3,11)(4,14)(5,9)(8,10)(18,24)(19,20)(21,30)(31,32)(34,39)(36,41)(37,46)(42,48);(1,3)(2,8)(4,15)(5,13)(6,11)(7,10)(9,16)(12,14)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,34)(35,41)(36,43)(37,40)(38,39)(42,47)(44,48)(45,46);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,36,38,41)(34,35,39,43)(37,44,46,47)(40,42,45,48);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,37,38,46)(34,40,39,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965965);