#8767442

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

4525 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8674638);