#421840

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

873 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,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, 329036);