#424649

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1222 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,e^d=ei,g^d=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, 331845);