#9059301

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,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, 8966497);