#8814003

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	#49910	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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)(4,10)(5,6)(7,9)(11,16)(12,13)(14,15)(18,21)(23,26)(24,29)(30,32)(33,34,38,39)(35,41,44,40)(36,37,45,46)(42,47,48,43);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(20,28)(24,29)(25,31)(30,32)(34,39)(36,45)(40,41)(42,48);(2,9)(3,4)(5,11)(13,15)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,31,32)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,46)(47,48);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,19,22,27)(18,23,21,26)(20,25,28,31)(24,30,29,32)(33,35,38,44)(34,40,39,41)(36,42,45,48)(37,43,46,47);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,20,22,28)(18,24,21,29)(19,25,27,31)(23,30,26,32)(33,36)(34,37)(35,42)(38,45)(39,46)(40,43)(41,47)(44,48);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,21,22,18)(19,26,27,23)(20,29,28,24)(25,32,31,30)(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=hi,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721199);