#8933161

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

4515 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:

(1,2,4,6)(3,7,8,5)(10,14)(15,17)(16,21)(22,23)(25,26,30,32)(27,31,35,40)(28,36,38,33)(29,39,37,34);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,27,30,35)(26,31,32,40)(28,37,38,29)(33,39,36,34);(1,2)(3,7)(4,6)(5,8)(11,18)(12,20)(15,17)(16,21)(26,32)(27,35)(28,38)(34,39);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,30,38)(26,33,32,36)(27,29,35,37)(31,34,40,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,29)(26,34)(27,28)(30,37)(31,33)(32,39)(35,38)(36,40);(1,3,4,8)(2,5,6,7)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(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=h,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, 8840357);