#8751420

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
prime radical	#44938	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gi,e^2=g,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, 8658616);