#8933169

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

4491 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)(3,8)(4,5)(6,7)(9,10,14,16)(11,13,19,15)(12,17,21,24)(18,22,20,23)(25,26,30,32)(27,31,35,40)(28,36,38,33)(29,39,37,34);(1,3,6,5)(2,4,7,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,22)(18,21)(23,24)(25,27,30,35)(26,31,32,40)(28,37,38,29)(33,39,36,34);(1,4)(2,5)(3,7)(6,8)(10,16)(12,21)(13,15)(18,20)(26,32)(27,35)(28,38)(34,39);(1,4,6,8)(2,5,7,3)(9,12)(10,17)(11,18)(13,22)(14,21)(15,23)(16,24)(19,20)(25,28,30,38)(26,33,32,36)(27,29,35,37)(31,34,40,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,22)(18,21)(23,24)(25,29)(26,34)(27,28)(30,37)(31,33)(32,39)(35,38)(36,40);(1,5,6,3)(2,4,7,8)(9,13,14,15)(10,11,16,19)(12,22,21,23)(17,18,24,20)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840365);