#8933092

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

4577 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,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,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, 8840288);