#10069076

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=gi,e^2=i,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, 9976272);