#8943940

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

4194 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)(3,10)(4,9)(5,14)(6,11)(7,12)(8,15)(13,16)(17,18,23,25)(19,26,20,24)(21,22,32,31)(27,30,29,28)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,3)(2,8)(4,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,18,23,25)(19,26,20,24)(21,31,32,22)(27,28,29,30)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,4,7,5)(2,9,12,14)(3,11,15,13)(6,8,16,10)(17,19,23,20)(18,24,25,26)(21,29,32,27)(22,30,31,28)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,5,7,4)(2,9,12,14)(3,13,15,11)(6,8,16,10)(17,20,23,19)(18,24,25,26)(21,27,32,29)(22,30,31,28)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,3)(2,10)(4,13)(5,11)(6,14)(7,15)(8,12)(9,16)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,6,7,16)(2,11,12,13)(3,14,15,9)(4,10,5,8)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851136);