#8271133

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
prime radical	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=ghi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8178329);