#10047385

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	#46532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gi,f^2=gh,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, 9954581);