#8659357

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

4539 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)(11,17)(12,16)(13,22)(14,18)(15,21)(19,24)(20,23)(25,26,31,35)(27,36,30,32)(28,34,29,33)(37,39,38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,31,30)(26,32,35,36)(28,37,29,38)(33,39,34,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,22)(11,19,21,20)(15,23,17,24)(25,28,31,29)(26,33,35,34)(27,37,30,38)(32,39,36,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,12)(10,16,18,22)(11,20,21,19)(15,23,17,24)(25,29,31,28)(26,34,35,33)(27,37,30,38)(32,39,36,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,17)(12,19)(13,20)(14,21)(15,18)(16,24)(22,23)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,39)(38,40);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,22)(14,18)(17,21)(19,23)(20,24)(25,30,31,27)(26,32,35,36)(28,38,29,37)(33,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8566553);