#8898324

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

4741 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:

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

Polycyclic presentation:

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

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, 8805520);