#8528945

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

4366 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:

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

Polycyclic presentation:

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

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, 8436141);