#9016616

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

4249 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=ghi,e^2=gi,f^2=gh,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, 8923812);