#8678922

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

4614 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=g,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, 8586118);