#396456

order	512
nilpotency class	3
isoclinism	isoclinic class of #394127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1252 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=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, 303652);