#8467213

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

4441 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)(3,4,8,7)(10,13)(11,12)(18,21)(19,20)(26,31)(28,32)(34,37)(36,39);(2,6)(4,7)(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3)(2,7)(4,6)(5,8)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,4,5,7)(2,3,6,8)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(9,13,15,10)(11,17,12,16)(14,21,22,18)(19,24,20,23)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,3)(2,4)(5,8)(6,7)(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8374409);