#8944470

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	#40920	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4190 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,30)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(2,6)(7,9)(8,12)(14,15)(17,19,22,25)(18,23,21,27)(20,26,30,28)(24,31,29,32)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,6)(4,13)(7,9)(10,16)(18,21)(19,25)(24,29)(26,28)(33,35,39,36)(34,41,38,40)(37,44,46,43)(42,47,45,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,30)(29,31)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,22,18)(19,27,25,23)(20,29,30,24)(26,32,28,31)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851666);