#8891790

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

4408 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8798986);