#10087795

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
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4208 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=h,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, 9994991);