#8517615

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	#40594	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4579 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8424811);