#8602056

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

4417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=gh,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, 8509252);