#8612202

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

4444 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^2=i,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, 8519398);