#9007315

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,e^2=gi,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, 8914511);