#430336

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1145 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)(11,22)(12,18)(13,23)(14,36)(15,20)(16,21)(17,25)(19,35)(24,37)(26,38)(27,29)(28,34)(30,39)(31,32)(33,40);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,28)(13,32)(14,26)(15,25)(16,27)(18,35)(19,37)(20,29)(21,38)(22,36)(23,30)(24,33)(31,40)(34,39);(2,6)(5,7)(9,12,16,31)(10,18,21,32)(11,23,27,19)(13,29,35,22)(14,30,15,37)(17,34,38,33)(20,24,36,39)(25,28,26,40);(3,8)(5,7)(9,13,14,33,16,35,15,34)(10,19,20,40,21,23,36,28)(11,24,25,32,27,39,26,18)(12,29,30,17,31,22,37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,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, 337532);