#8418405

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

4015 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=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, 8325601);