#8898329

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

4532 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)(11,18)(12,17)(13,19)(14,24)(15,16)(20,23)(21,22)(25,26,30,34)(27,35,36,31)(28,32,37,40)(29,33,38,39);(3,7)(6,8)(10,16)(13,24)(17,18)(20,22)(27,36)(29,38)(31,35)(33,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,18)(13,20,24,22)(14,23,19,21)(25,27,30,36)(26,31,34,35)(28,29,37,38)(32,39,40,33);(9,12,15,11)(10,18,16,17)(13,20,24,22)(14,23,19,21)(25,28)(26,32)(27,29)(30,37)(31,39)(33,35)(34,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,13)(10,19)(11,20)(12,22)(14,16)(15,24)(17,21)(18,23)(25,29)(26,33)(27,37)(28,36)(30,38)(31,32)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,19)(10,13,16,24)(11,21,12,23)(17,22,18,20)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,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, 8805525);