#430334

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1201 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)(11,20)(12,26)(13,31)(14,18)(15,19)(16,23)(17,30)(21,25)(22,34)(24,33)(27,32)(28,35)(29,39)(36,38)(37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,21)(19,33)(20,31)(22,28)(26,36)(29,38)(30,37)(32,35)(39,40);(2,6)(5,7)(9,10)(11,21)(12,17)(13,18)(14,31)(15,19)(16,23)(20,25)(22,36)(24,33)(26,30)(27,32)(28,35)(29,39)(34,38)(37,40);(3,8)(5,7)(9,12,13,28,15,30,14,29)(10,17,18,35,19,26,31,39)(11,22,23,27,25,38,24,37)(16,32,20,34,33,40,21,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=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, 337530);