#10143441

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 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:

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

Polycyclic presentation:

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

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, 10050637);