#396733

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,h^d=hi

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, 303929);