#9016477

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
prime radical	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=ghi,e^2=i,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, 8923673);