#416619

order	512
nilpotency class	3
isoclinism	isoclinic class of #397950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1756 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)(4,5)(6,8)(9,10)(11,21)(12,25)(13,20)(14,19)(15,18)(16,23)(17,28)(22,37)(24,35)(26,31)(27,33)(29,34)(30,32)(36,38)(39,40);(2,5)(6,7)(9,11,14,30,15,24,13,23)(10,16,20,35,18,32,19,21)(12,26,28,38,29,22,27,40)(17,31,25,39,33,37,34,36);(9,12,14,28,15,29,13,27)(10,17,20,25,18,33,19,34)(11,22,23,38,24,26,30,40)(16,31,21,36,32,37,35,39);(1,3,4,8)(2,6,5,7)(10,18)(16,32)(17,33)(19,20)(21,35)(25,34)(31,37)(36,39)

Polycyclic presentation:

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

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, 323815);