#8927986

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

4417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8835182);