#8813973

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	#43255	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721169);