#8718469

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

4467 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gi,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, 8625665);