#8907963

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

4068 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,47,44);(1,3,6,11)(2,7,4,8)(5,10,13,16)(9,14,12,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,27)(25,32)(26,31)(33,35)(34,39)(36,44)(37,40)(38,43)(41,45)(42,46)(47,48);(2,4)(5,13)(7,8)(10,16)(18,22)(19,27)(20,30)(26,32)(35,43)(36,45)(39,46)(40,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20,23,30)(18,25,22,31)(19,28,27,21)(24,32,29,26)(33,36)(34,40)(35,44)(37,39)(38,45)(41,43)(42,48)(46,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21,23,28)(18,26,22,32)(19,20,27,30)(24,25,29,31)(33,37)(34,41)(35,40)(36,46)(38,47)(39,45)(42,44)(43,48);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,22,23,18)(19,29,27,24)(20,31,30,25)(21,32,28,26)(33,35)(34,39)(36,44)(37,40)(38,43)(41,45)(42,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815159);