#424582

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1278 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)(11,15)(12,22)(13,25)(14,17)(16,24)(18,27)(19,33)(20,28)(21,37)(23,32)(26,31)(29,35)(30,36)(34,40)(38,39);(12,23)(15,28)(16,31)(22,32)(24,26)(29,39)(30,37)(33,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,29)(17,28)(18,30)(22,33)(23,34)(24,35)(25,37)(26,38)(27,36)(31,39)(32,40);(9,12)(10,16)(11,19)(13,26)(14,23)(15,29)(17,31)(18,32)(20,34)(21,38)(22,25)(24,27)(28,39)(30,40)(33,37)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331778);