#10040814

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3885 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,21)(19,25,27,20)(22,29,30,24)(26,28,32,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,27)(18,20,21,25)(22,26,30,32)(24,28,29,31)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(2,3)(7,9)(8,10)(14,15)(18,21)(20,25)(24,29)(28,31)(33,35,37,42)(34,38,40,46)(36,41,44,43)(39,45,48,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,18)(19,20,27,25)(22,24,30,29)(26,31,32,28)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,30)(18,24,21,29)(19,26,27,32)(20,28,25,31)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9948010);