#8471041

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

4515 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=ghi,e^2=hi,f^2=gi,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, 8378237);