#396606

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,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, 303802);