#8260854

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	#52278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=g,e^2=gi,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

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, 8168050);