#8862385

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

4605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8769581);