#10068998

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

4204 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(19,25)(23,26)(27,31)(30,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,2,5,8)(3,6,9,13)(4,7,12,16)(10,14,11,15)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(35,41)(38,45)(42,47)(46,48);(3,9)(6,13)(10,11)(14,15)(18,20)(23,26)(24,28)(30,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(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);(1,3,5,9)(2,6,8,13)(4,11,12,10)(7,15,16,14)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=i,e^2=gi,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, 9976194);