#8108316

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

4626 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015512);