#8215260

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	#56800	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8122456);