#9007890

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	#2424	C₂²
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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8915086);