#8726660

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

4098 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)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(33,34,40,45)(35,41,47,48)(36,44,38,42)(37,43,46,39);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,14)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,29)(31,32)(33,35,40,47)(34,41,45,48)(36,37,38,46)(39,44,43,42);(4,13)(7,12)(11,16)(14,15)(18,23)(19,25)(26,32)(27,29)(33,36,40,38)(34,42,45,44)(35,46,47,37)(39,41,43,48);(1,4,6,13)(2,7,10,12)(3,11,9,16)(5,15,8,14)(17,19,22,25)(18,24,23,20)(21,29,31,27)(26,28,32,30)(33,37,40,46)(34,43,45,39)(35,36,47,38)(41,44,48,42);(1,3)(2,8)(4,11)(5,10)(6,9)(7,14)(12,15)(13,16)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,32,29)(33,38,40,36)(34,44,45,42)(35,46,47,37)(39,41,43,48);(1,5,6,8)(2,9,10,3)(4,14,13,15)(7,11,12,16)(17,21,22,31)(18,26,23,32)(19,27,25,29)(20,28,24,30)(33,39,40,43)(34,37,45,46)(35,44,47,42)(36,41,38,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633856);