#8429526

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336722);