#8701384

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	#59110	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4448 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608580);