#8214210

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	#41495	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=i,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, 8121406);