#437972

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^c=hi,f^d=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, 345168);