#8936275

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

4382 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=g,f^2=h,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, 8843471);