#8505733

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

4388 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8412929);