#8528046

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

4381 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)(9,10,14,17)(11,18,12,15)(13,16,21,24)(19,23,20,22)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(3,7)(6,8)(11,12)(15,18)(19,20)(22,23)(26,28)(27,33)(32,36)(35,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,17,18)(13,19,21,20)(16,22,24,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,15,17,18)(13,19,21,20)(16,23,24,22)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,24)(18,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8435242);