#8924369

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

4506 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)(11,19)(12,15)(13,17)(14,22)(16,20)(18,24)(21,23)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(3,7)(6,8)(9,11,15,20)(10,16,12,19)(13,22,23,18)(14,17,24,21)(27,29)(31,34)(35,36)(38,39);(1,2)(3,6)(4,5)(7,8)(10,12)(14,24)(16,19)(17,21)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(9,12,15,10)(11,19,20,16)(13,17,23,21)(14,22,24,18)(25,28,30,37)(26,32,33,40)(27,35,29,36)(31,38,34,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,23)(10,17,12,21)(11,18,20,22)(14,19,24,16)(25,29,30,27)(26,31,33,34)(28,35,37,36)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,14,15,24)(10,18,12,22)(11,17,20,21)(13,19,23,16)(25,26)(27,34)(28,32)(29,31)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831565);