#10171800

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=hi,e^2=gi,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, 10078996);