#8197931

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

4659 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:

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

Polycyclic presentation:

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

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, 8105127);