#8603509

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

4559 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=h,e^2=h,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, 8510705);