#8614119

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

4442 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)(11,17)(12,22)(13,18)(14,16)(15,19)(20,23)(21,24)(26,31)(27,33)(28,35)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,16,19)(10,17,14,15)(12,21,13,20)(18,23,22,24)(25,26,30,31)(27,29,33,39)(28,36,35,32)(34,40,37,38);(2,3)(6,7)(9,12,16,13)(10,18,14,22)(11,20,19,21)(15,23,17,24)(25,27,30,33)(26,29,31,39)(28,37,35,34)(32,40,36,38);(4,8)(6,7)(9,13,16,12)(10,18,14,22)(11,21,19,20)(15,23,17,24)(25,26)(27,29)(28,32)(30,31)(33,39)(34,38)(35,36)(37,40);(1,3,5,2)(4,7,8,6)(9,14,16,10)(11,17,19,15)(12,18,13,22)(20,23,21,24)(25,28,30,35)(26,32,31,36)(27,34,33,37)(29,38,39,40);(1,4)(2,6)(3,7)(5,8)(9,15,16,17)(10,19,14,11)(12,23,13,24)(18,21,22,20)(25,29)(26,33)(27,31)(28,38)(30,39)(32,37)(34,36)(35,40)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gh,e^2=gi,f^2=gh,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, 8521315);