#8688303

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

4555 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595499);