#9017311

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	#46962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(3,5)(4,9)(6,16)(7,10)(8,14)(11,13)(12,15)(33,34,38,39)(35,41,44,40)(36,37,45,46)(42,47,48,43);(1,3,7,6)(2,5,10,16)(4,12,14,11)(8,13,9,15)(17,18,23,27)(19,28,30,24)(20,22,26,32)(21,25,29,31)(34,39)(36,45)(40,41)(42,48);(1,4,7,14)(2,8,10,9)(3,11,6,12)(5,15,16,13)(17,19,23,30)(18,24,27,28)(20,21,26,29)(22,31,32,25)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,46)(47,48);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,10)(13,16)(17,20,23,26)(18,22,27,32)(19,29,30,21)(24,25,28,31)(33,35,38,44)(34,40,39,41)(36,42,45,48)(37,43,46,47);(1,5)(2,3)(4,13)(6,10)(7,16)(8,12)(9,11)(14,15)(17,21,23,29)(18,25,27,31)(19,20,30,26)(22,24,32,28)(33,36)(34,37)(35,42)(38,45)(39,46)(40,43)(41,47)(44,48);(1,6,7,3)(2,5,10,16)(4,11,14,12)(8,13,9,15)(17,22)(18,26)(19,25)(20,27)(21,24)(23,32)(28,29)(30,31)(33,37)(34,36)(35,43)(38,46)(39,45)(40,42)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8924507);