#8934461

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

4453 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,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,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, 8841657);