#8944478

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	#41030	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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)(3,4)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(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);(2,7)(4,8)(9,13)(12,15)(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);(2,7)(4,8)(5,14)(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);(1,3,6,11)(2,8,7,4)(5,10,14,16)(9,15,13,12)(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);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,30)(29,31)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,13)(8,15)(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,e^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, 8851674);