#8892553

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

4649 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8799749);