#8716886

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

4465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=gi,e^2=gh,f^2=g,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, 8624082);