#421762

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1485 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)(9,10,14,19,16,13,15,30)(11,17,22,33,23,21,24,38)(12,25,28,27,29,35,31,18)(20,36,37,34,26,40,39,32);(1,3,4,8)(2,5,6,7)(9,11,14,22,16,23,15,24)(10,17,19,33,13,21,30,38)(12,26,28,39,29,20,31,37)(18,34,25,40,27,32,35,36);(9,12)(10,18)(11,20)(13,27)(14,31)(15,28)(16,29)(17,32)(19,35)(21,34)(22,39)(23,26)(24,37)(25,30)(33,40)(36,38);(3,8)(5,7)(9,13,16,10)(11,21,23,17)(12,27,29,18)(14,30,15,19)(20,34,26,32)(22,38,24,33)(25,28,35,31)(36,37,40,39)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,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, 328958);