#8908519

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

4424 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815715);