#8161281

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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)(3,12)(4,8)(5,13)(6,11)(7,14)(9,15)(10,16)(17,18)(19,25)(20,21)(22,30)(23,24)(26,31)(27,28)(29,32)(37,45)(40,47)(43,46)(44,48);(1,3)(2,7)(4,13)(5,9)(6,14)(8,16)(10,15)(11,12)(18,24)(20,25)(26,30)(28,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,4,6,15)(2,8,11,9)(3,13,14,10)(5,7,16,12)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,30,32)(33,35,38,39)(34,36,41,42)(37,43,45,46)(40,44,47,48);(1,4)(2,9)(3,13)(5,7)(6,15)(8,11)(10,14)(12,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32)(33,36,38,42)(34,39,41,35)(37,44,45,48)(40,46,47,43);(1,2)(3,7)(4,8)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5,6,16)(2,10,11,13)(3,9,14,8)(4,7,15,12)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8068477);