#469148

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

861 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)(6,7)(9,10,15,20)(11,17,23,34)(12,24,22,16)(13,18)(14,31,25,19)(21,38)(26,32)(27,33)(28,36)(29,35)(30,37)(39,40);(1,2,4,5)(3,7,8,6)(9,11,14,22,15,23,25,12)(10,16,19,17,20,24,31,34)(13,21,29,39,30,27,28,26)(18,32,36,33,37,40,35,38);(1,3)(2,6)(4,8)(5,7)(9,12,25,23,15,22,14,11)(10,17,31,16,20,34,19,24)(13,27,29,26,30,21,28,39)(18,32,36,33,37,40,35,38);(3,8)(6,7)(9,13,14,29,15,30,25,28)(10,18,19,36,20,37,31,35)(11,21,22,39,23,27,12,26)(16,32,17,33,24,40,34,38)

Polycyclic presentation:

2^9|a^2=efg,b^2=fg,c^2=egi,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,h^c=hi,e^d=ei,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, 376344);