#10166245

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	#48577	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18)(19,27)(20,28)(21,25)(22,30)(23,24)(26,29)(31,32)(33,34,39,43)(35,38,42,40)(36,37,41,48)(44,46,47,45);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19,23,28)(18,20,24,27)(21,29,31,22)(25,30,32,26)(33,35,39,42)(34,40,43,38)(36,44,41,47)(37,45,48,46);(2,9)(3,4)(10,14)(11,12)(18,24)(20,27)(25,32)(26,30)(33,36,39,41)(34,37,43,48)(35,44,42,47)(38,46,40,45);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,20,23,27)(18,19,24,28)(21,30,31,26)(22,25,29,32)(33,34)(35,40)(36,37)(38,42)(39,43)(41,48)(44,45)(46,47);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,21,23,31)(18,25,24,32)(19,22,28,29)(20,26,27,30)(33,37)(34,41)(35,45)(36,43)(38,44)(39,48)(40,47)(42,46);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,22)(18,26)(19,21)(20,25)(23,29)(24,30)(27,32)(28,31)(33,38,39,40)(34,42,43,35)(36,46,41,45)(37,47,48,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=h,e^2=hi,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, 10073441);