#9245122

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(3,8)(4,5)(6,7)(9,13)(10,11)(12,14)(15,16)(17,18)(19,28)(20,30)(21,25)(22,29)(23,24)(26,31)(27,32)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(2,7)(5,9)(8,14)(11,15)(18,24)(19,21)(26,30)(27,29)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,3,6,12)(2,8,7,14)(4,10,13,16)(5,11,9,15)(17,19,23,21)(18,25,24,28)(20,27,31,29)(22,26,32,30)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,4)(2,9)(3,10)(5,7)(6,13)(8,15)(11,14)(12,16)(17,20,23,31)(18,26,24,30)(19,29,21,27)(22,28,32,25);(1,3)(2,8)(4,10)(5,11)(6,12)(7,14)(9,15)(13,16)(17,21,23,19)(18,25,24,28)(20,27,31,29)(22,30,32,26)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,5,6,9)(2,4,7,13)(3,11,12,15)(8,10,14,16)(17,22)(18,27)(19,26)(20,28)(21,30)(23,32)(24,29)(25,31)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9152318);