#9016488

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

4130 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,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, 8923684);