#474326

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

636 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)(9,10,14,20,16,22,15,21)(11,23,26,35,27,36,30,17)(12,18,31,13,28,32,19,29)(24,25,34,33,39,37,38,40);(2,5)(6,8)(9,11,15,26,16,27,14,30)(10,17,21,23,22,35,20,36)(12,24,31,38,28,39,19,34)(13,25,32,40,29,37,18,33);(3,7)(6,8)(9,12)(10,18)(11,24,27,39)(13,20)(14,31)(15,19)(16,28)(17,33,35,40)(21,29)(22,32)(23,37,36,25)(26,34,30,38);(1,3,4,7)(2,6,5,8)(9,13,14,18,16,29,15,32)(10,19,20,28,22,31,21,12)(11,25,30,33,27,37,26,40)(17,34,36,39,35,38,23,24)

Polycyclic presentation:

2^9|a^2=efg,b^2=fgh,c^2=f,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^c=gi,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, 381522);