#8614137

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

4675 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8521333);