#8936727

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

4459 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)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26)(27,35)(28,32)(29,33)(30,40)(31,34)(36,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,17)(14,19)(16,24)(18,21)(22,23)(25,26)(27,35)(28,32)(29,30)(31,34)(33,40)(36,39)(37,38);(2,5)(3,7)(10,13)(12,21)(15,17)(18,20)(25,27,31,28)(26,32,34,35)(29,38,40,36)(30,39,33,37);(3,7)(6,8)(9,11)(10,15)(12,20)(13,17)(14,19)(16,24)(18,21)(22,23)(25,28,31,27)(26,32,34,35)(29,36,40,38)(30,39,33,37);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,16,13,22)(11,18,19,20)(15,23,17,24)(25,29)(26,33)(27,36)(28,38)(30,34)(31,40)(32,39)(35,37);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,17,19,15)(12,22,21,16)(18,24,20,23)(25,30,31,33)(26,29,34,40)(27,37,28,39)(32,36,35,38)

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8843923);