#8107416

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

4622 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=i,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,d^c=dgi

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, 8014612);