#8096964

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

4654 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004160);