#8726823

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	#41120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4268 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)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(3,7)(5,8)(10,15)(11,19)(17,20)(18,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,19)(17,20)(18,24)(27,34)(30,31)(35,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,19)(10,16,15,21)(12,18,22,24)(13,17,23,20)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,22)(10,17,15,20)(11,18,19,24)(13,16,23,21)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(9,13,14,23)(10,18,15,24)(11,20,19,17)(12,21,22,16)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40)

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=g,f^2=g,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, 8634019);