#8994517

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

4212 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,23,21)(19,20,27,24)(22,29,30,25)(26,31,32,28)(33,34,37,41)(35,39,43,38)(36,40,45,44)(42,47,48,46);(1,3)(2,8)(4,13)(5,7)(6,10)(9,11)(12,14)(15,16)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,9)(3,5)(4,14)(6,16)(18,21)(20,24)(22,30)(26,32)(33,34,37,41)(35,38,43,39)(36,44,45,40)(42,47,48,46);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,15,16,10)(17,21,23,18)(19,24,27,20)(22,29,30,25)(26,31,32,28)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,22,23,30)(18,25,21,29)(19,26,27,32)(20,28,24,31)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=i,e^2=h,f^2=g,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, 8901713);