#8214515

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

4420 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8121711);