#8907989

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

4611 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815185);