#8291638

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

3874 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,6,5)(2,7,8,10)(4,13,14,11)(9,16,12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(34,39)(35,40)(41,46)(42,47);(2,8)(7,10)(9,12)(15,16)(19,25)(23,26)(27,31)(30,32)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,4,6,14)(2,9,8,12)(3,11,5,13)(7,15,10,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,5,6,3)(2,10,8,7)(4,11,14,13)(9,15,12,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198834);