#8829026

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

4428 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8736222);