#8471136

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

4587 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gi,e^2=gi,f^2=ghi,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, 8378332);