#8509013

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

4504 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=gi,e^2=i,f^2=hi,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, 8416209);