#10042545

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	#44112	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=gi,e^2=g,f^2=h,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, 9949741);