#8936679

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

4498 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)(11,18)(12,17)(13,22)(14,19)(15,20)(16,23)(21,24)(25,26,29,33)(27,31,35,30)(28,36,38,32)(34,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,27,29,35)(26,30,33,31)(28,34,38,37)(32,39,36,40);(2,5)(3,7)(9,12,15,13)(10,17,20,22)(11,19,23,21)(14,16,24,18)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(3,7)(6,8)(9,13,15,12)(10,17,20,22)(11,21,23,19)(14,16,24,18)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,18)(12,21)(13,19)(14,22)(15,23)(16,20)(17,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,14,15,24)(10,19,20,21)(11,22,23,17)(12,18,13,16)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8843875);