#8383486

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8290682);