#8508906

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

4393 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,18)(13,17)(14,16)(15,19)(21,23)(22,24)(25,26,31,35)(27,36,30,32)(28,33,34,29)(37,40,39,38);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,19,20)(12,21,13,22)(17,23,18,24)(25,27,31,30)(26,32,35,36)(28,37,34,39)(29,40,33,38);(1,2)(3,5)(4,6)(7,8)(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24)(25,28,31,34)(26,33,35,29)(27,37,30,39)(32,38,36,40);(1,3,4,8)(2,5,6,7)(9,13,15,12)(10,18,19,17)(11,21,14,22)(16,23,20,24)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,38)(39,40);(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,29,31,33)(26,34,35,28)(27,38,30,40)(32,37,36,39);(9,14,15,11)(10,16,19,20)(12,22,13,21)(17,23,18,24)(25,30,31,27)(26,32,35,36)(28,39,34,37)(29,40,33,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8416102);