#8678459

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=gh,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, 8585655);