#8750204

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

4436 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657400);