#10143412

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	#54758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3923 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,26)(20,25)(21,22)(23,27)(24,31)(28,30)(29,32)(33,34,39,37)(35,40,45,47)(36,42,46,41)(38,48,44,43);(2,7)(4,10)(9,13)(12,14)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,35,39,45)(34,40,37,47)(36,44,46,38)(41,48,42,43);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,20,23,21)(18,25,27,22)(19,28,31,29)(24,32,26,30)(33,36,39,46)(34,41,37,42)(35,38,45,44)(40,43,47,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,36)(34,42)(35,44)(37,41)(38,45)(39,46)(40,43)(47,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,22,23,25)(18,21,27,20)(19,30,31,32)(24,29,26,28)(33,37,39,34)(35,40,45,47)(36,42,46,41)(38,43,44,48);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,19)(18,26)(20,28)(21,29)(22,32)(23,31)(24,27)(25,30)(33,38,39,44)(34,43,37,48)(35,36,45,46)(40,41,47,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=ghi,e^2=gh,f^2=h,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, 10050608);