#8323149

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

4430 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8230345);