#149463

order	512
nilpotency class	4
isoclinism	isoclinic class of #129559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dh,h^b=hi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56659);