#8932419

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

4390 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839615);