#8436638

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

4471 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=gh,f^2=g,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, 8343834);