#8680004

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

4373 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587200);