#427864

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,25)(13,30)(14,31)(15,20)(16,19)(17,23)(18,29)(22,34)(24,37)(26,40)(27,33)(28,35)(32,36)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,26)(13,14)(15,24)(16,23)(18,34)(19,37)(20,30)(21,31)(22,28)(25,38)(27,29)(32,39)(33,36)(35,40);(2,6)(5,7)(9,12)(10,18)(11,22)(13,27)(14,28)(15,29)(16,32)(17,33)(19,35)(20,25)(21,38)(23,26)(24,39)(30,40)(31,36)(34,37);(3,8)(5,7)(9,13,16,24,15,23,14,11)(10,17,31,37,20,30,19,21)(12,27,32,39,29,26,28,22)(18,33,36,34,25,40,35,38)

Polycyclic presentation:

2^9|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, 335060);