#8943982

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

4227 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 8851178);