#8932630

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

4569 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)(3,5,7,8)(11,17)(12,20)(15,18)(16,21)(25,26,31,34)(27,28,32,29)(30,33,38,40)(35,36,39,37);(3,7)(5,8)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,27,31,32)(26,29,34,28)(30,35,38,39)(33,37,40,36);(1,2)(3,5)(4,6)(7,8)(10,14)(15,18)(16,21)(23,24)(25,28,31,29)(26,32,34,27)(30,37,38,36)(33,35,40,39);(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29,31,28)(26,32,34,27)(30,37,38,36)(33,39,40,35);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30,31,38)(26,33,34,40)(27,35,32,39)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^2=g,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, 8839826);