#149464

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×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,g^2=i,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, 56660);