#8159503

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

4080 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

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, 8066699);