#8701413

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

4464 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608609);