#8930530

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

4355 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837726);