#467785

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

829 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fgi,c^2=e,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, 374981);