#8479531

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

4182 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8386727);