#8496478

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

4590 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8403674);