#8932868

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
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^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,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, 8840064);