#8701370

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

4582 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608566);