#8383462

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4448 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8290658);