#8873724

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	#39262	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780920);