#8211615

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	#44131	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8118811);