#8943965

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	#43478	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=h,e^2=g,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, 8851161);