#8970217

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

4478 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8877413);