#8898336

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

4531 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)(9,10,16,20)(11,21,24,17)(12,15,14,18)(13,23,22,19)(25,26,30,34)(27,35,36,31)(28,32,37,40)(29,33,38,39);(3,7)(6,8)(9,11,16,24)(10,17,20,21)(12,22,14,13)(15,23,18,19)(27,36)(29,38)(31,35)(33,39);(1,2)(3,6)(4,5)(7,8)(9,12,16,14)(10,18,20,15)(11,22,24,13)(17,19,21,23)(25,27,30,36)(26,31,34,35)(28,29,37,38)(32,39,40,33);(9,13)(10,19)(11,12)(14,24)(15,17)(16,22)(18,21)(20,23)(25,28)(26,32)(27,29)(30,37)(31,39)(33,35)(34,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,14,16,12)(10,18,20,15)(11,22,24,13)(17,23,21,19)(25,29)(26,33)(27,37)(28,36)(30,38)(31,32)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,15,16,18)(10,12,20,14)(11,23,24,19)(13,17,22,21)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,e^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, 8805532);