#8718968

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

4682 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8626164);