#8093498

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

4127 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)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(18,21)(19,25)(24,28)(26,31)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(3,11)(4,13)(8,14)(9,16)(17,18,22,21)(19,23,25,27)(20,28,29,24)(26,32,31,30)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(2,7)(3,11)(9,16)(10,12)(19,25)(23,27)(26,31)(30,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,19,22,25)(18,23,21,27)(20,26,29,31)(24,30,28,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(17,20,22,29)(18,24,21,28)(19,26,25,31)(23,30,27,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,21,22,18)(19,27,25,23)(20,28,29,24)(26,32,31,30)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000694);