#10166167

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	#51441	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=gh,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, 10073363);