#8417312

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

4367 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324508);