#10056599

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=h,e^2=ghi,f^2=g,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, 9963795);