#10087308

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

4283 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)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,40)(36,37);(1,2)(3,6)(4,7)(5,8)(10,15)(12,17)(16,21)(20,22)(25,27,31,29)(26,28,34,32)(30,35,39,37)(33,36,38,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(16,21)(19,24)(25,28,31,32)(26,29,34,27)(30,36,39,40)(33,37,38,35);(1,2)(3,5)(4,7)(6,8)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,29,31,27)(26,32,34,28)(30,37,39,35)(33,40,38,36);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,26)(27,32)(28,29)(30,38)(31,34)(33,39)(35,36)(37,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 9994504);