#8924367

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

4669 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,e^2=gi,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, 8831563);