#8464128

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

4585 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)(11,20)(12,16)(13,17)(14,18)(15,19)(21,22)(23,24)(25,26,32,30)(27,37,38,33)(28,35,29,34)(31,36,40,39);(3,7)(6,8)(11,12)(16,20)(21,23)(22,24)(25,27)(26,33)(28,36)(29,39)(30,37)(31,35)(32,38)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,19,20)(13,21,18,23)(14,22,17,24)(25,28,32,29)(26,34,30,35)(27,36,38,39)(31,33,40,37);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,16,19,20)(13,23,18,21)(14,22,17,24)(25,29,32,28)(26,35,30,34)(27,36,38,39)(31,33,40,37);(1,2)(3,6)(4,5)(7,8)(9,13,15,18)(10,17,19,14)(11,21,12,23)(16,24,20,22)(25,30,32,26)(27,37,38,33)(28,35,29,34)(31,39,40,36);(9,14,15,17)(10,18,19,13)(11,22,12,24)(16,23,20,21)(25,31)(26,36)(27,35)(28,33)(29,37)(30,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8371324);