#474259

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

614 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381455);