#8096982

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

4459 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004178);