#8891717

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

4434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gh,f^2=g,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, 8798913);