#8902505

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

4585 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,13)(14,15)(16,21)(17,23)(19,20)(22,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2,5,6)(3,7,8,4)(10,15)(12,17)(16,18)(19,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3)(2,4)(5,8)(6,7)(10,15)(13,23)(16,18)(20,24)(28,36)(31,38)(34,40)(37,39);(1,4)(2,3)(5,7)(6,8)(9,11,14,21)(10,16,15,18)(12,22,17,19)(13,20,23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,4)(5,6)(7,8)(9,12,14,17)(10,13,15,23)(11,19,21,22)(16,24,18,20)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,20)(12,15)(14,23)(16,19)(18,22)(21,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=hi,e^2=gi,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, 8809701);