#8924667

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

4322 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)(3,7)(4,8)(5,6)(9,10,15,17)(11,16,21,24)(12,18,22,23)(13,19,20,14);(2,6)(3,8)(9,11,15,21)(10,16,17,24)(12,20,22,13)(14,23,19,18)(25,26,31,34)(27,35,29,32)(28,30,33,40)(36,39,38,37);(1,3)(2,4)(5,8)(6,7)(10,17)(11,21)(12,22)(14,19)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,39,40,37);(1,4)(2,3)(5,7)(6,8)(9,12)(10,18)(11,20)(13,21)(14,16)(15,22)(17,23)(19,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,37,35,39);(1,2)(3,4)(5,6)(7,8)(9,13,15,20)(10,19,17,14)(11,22,21,12)(16,23,24,18)(25,29,31,27)(26,32,34,35)(28,36,33,38)(30,39,40,37);(1,4)(2,3)(5,7)(6,8)(9,14,15,19)(10,20,17,13)(11,23,21,18)(12,16,22,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,38)(35,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831863);