#8621807

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

4495 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)(3,8)(4,5)(6,7)(9,10,15,18)(11,13,16,21)(12,14,17,23)(19,20,22,24)(25,26,32,30)(27,31,35,33)(28,34,29,39)(36,38,37,40);(1,3,6,5)(2,4,7,8)(9,11,15,16)(10,13,18,21)(12,19,17,22)(14,20,23,24)(25,27,32,35)(26,33,30,31)(28,36,29,37)(34,40,39,38);(2,7)(4,8)(11,16)(13,21)(19,22)(20,24)(25,28,32,29)(26,34,30,39)(27,36,35,37)(31,38,33,40);(2,7)(3,5)(9,12)(10,14)(11,19)(13,20)(15,17)(16,22)(18,23)(21,24)(25,29,32,28)(26,34,30,39)(27,37,35,36)(31,38,33,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,21)(10,16,18,11)(12,20,17,24)(14,22,23,19)(25,30,32,26)(27,33,35,31)(28,34,29,39)(36,38,37,40);(1,5,6,3)(2,4,7,8)(9,14)(10,17)(11,20)(12,18)(13,22)(15,23)(16,24)(19,21)(25,31,32,33)(26,35,30,27)(28,38,29,40)(34,37,39,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=gh,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,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529003);