#8421704

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

4698 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8328900);