#8180399

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	#42619	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087595);