#8174316

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

4584 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081512);