#8289376

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	#43101	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4435 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:

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

Polycyclic presentation:

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

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, 8196572);