#8922944

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

4338 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,22)(20,24)(21,23)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(2,6)(3,7)(9,11,15,19)(10,16,14,22)(12,20,21,13)(17,24,23,18)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3,5,7)(2,4,6,8)(10,14)(11,19)(12,21)(18,24)(27,33)(28,36)(31,35)(32,37);(1,4)(2,7)(3,6)(5,8)(9,12)(10,17)(11,20)(13,19)(14,23)(15,21)(16,24)(18,22)(25,28,30,36)(26,32,29,37)(27,34,33,39)(31,38,35,40);(9,13,15,20)(10,18,14,24)(11,21,19,12)(16,23,22,17)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40);(9,14,15,10)(11,22,19,16)(12,23,21,17)(13,24,20,18)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8830140);