#8720828

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

4571 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8628024);