#8429532

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,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, 8336728);