#8994410

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

4164 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)(6,14)(7,9)(12,16)(13,15)(18,23)(19,26)(25,30)(27,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,3,7,4)(2,8,9,10)(5,14,15,12)(6,13,16,11)(17,18,22,23)(19,24,26,20)(21,25,29,30)(27,31,32,28)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(2,9)(3,4)(5,15)(6,16)(19,26)(20,24)(21,29)(25,30)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=gi,e^2=gh,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, 8901606);