#10195909

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	#47088	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3934 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)(3,8)(4,6)(5,12)(7,11)(9,14)(10,15)(13,16)(17,18)(19,24)(20,22)(21,26)(23,25)(27,32)(28,31)(29,30)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,42,44,48);(1,3,6,11)(2,7,4,8)(5,10,14,13)(9,15,12,16)(17,19)(18,24)(20,29)(21,27)(22,30)(23,28)(25,31)(26,32)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,48,46,42);(2,4)(7,8)(9,12)(15,16)(18,25)(19,28)(20,27)(26,30)(33,34,39,37)(35,40,45,43)(36,41,47,46)(38,42,44,48);(1,4,6,2)(3,8,11,7)(5,12,14,9)(10,16,13,15)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,30,31,26)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,48)(42,43);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,21,23,29)(18,26,25,30)(19,20,28,27)(22,31,32,24)(33,37,39,34)(35,43,45,40)(36,41,47,46)(38,42,44,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,22)(18,27)(19,30)(20,25)(21,24)(23,32)(26,28)(29,31)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,41,43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=ghi,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, 10103105);