#10171679

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	#46555	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=ghi,e^2=gi,f^2=g,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, 10078875);