#8508920

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

4342 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)(3,7)(9,10,15,19)(11,20,12,16)(13,17,23,21)(14,18,24,22)(26,30)(31,36)(32,37)(38,40);(1,2,4,5)(3,6,7,8)(11,12)(13,23)(16,20)(17,21)(27,33)(31,36)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,19,20)(13,18,23,22)(14,21,24,17)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,16,19,20)(13,22,23,18)(14,21,24,17)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13,15,23)(10,17,19,21)(11,18,12,22)(14,20,24,16)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,14)(10,18)(11,21)(12,17)(13,20)(15,24)(16,23)(19,22)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416116);