#8096659

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

4588 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=h,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, 8003855);