#8533671

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

4809 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440867);