#10166124

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	#48241	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,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, 10073320);