#9059190

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(19,24)(20,26)(21,27)(22,29)(23,25)(28,32)(30,31);(2,7)(3,11)(4,14)(13,16)(17,19,23,21)(18,24,25,27)(20,28,30,29)(22,26,32,31)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(3,11)(4,14)(5,9)(8,10)(18,25)(19,21)(20,30)(22,32)(34,38)(35,42)(36,45)(47,48);(1,3)(2,8)(4,15)(5,13)(6,11)(7,10)(9,16)(12,14)(17,20,23,30)(18,26,25,31)(19,29,21,28)(22,27,32,24)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,21,23,19)(18,27,25,24)(20,29,30,28)(22,31,32,26)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22)(18,28)(19,26)(20,27)(21,31)(23,32)(24,30)(25,29)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^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, 8966386);