#8924355

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

4319 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=gi,e^2=i,f^2=g,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, 8831551);