#8750367

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	#42547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 8657563);