#10069264

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

4342 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=i,e^2=hi,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, 9976460);