#438073

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1029 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:

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

Polycyclic presentation:

2^9|a^2=e,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, 345269);