#8093486

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

3983 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)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,30)(31,32)(34,36)(39,41)(40,44)(46,47);(3,11)(4,13)(8,14)(9,16)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(3,11)(9,16)(10,12)(19,26)(22,25)(27,32)(30,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=h,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, 8000682);