#8325763

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

4479 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8232959);