#8933892

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

4420 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,e^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, 8841088);