#467132

order	512
nilpotency class	3
isoclinism	isoclinic class of #454970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1092 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)(9,10,14,17,13,18,27,33)(11,19,22,30,20,34,28,15)(12,23,26,38,25,16,31,32)(21,35,36,37,24,39,40,29);(1,3,4,7)(2,5,6,8)(11,20)(14,27)(17,33)(19,34)(21,24)(26,31)(32,38)(35,39);(3,7)(5,8)(9,11)(10,15)(12,24)(13,20)(14,28)(16,29)(17,34)(18,30)(19,33)(21,25)(22,27)(23,37)(26,36)(31,40)(32,39)(35,38);(9,12)(10,16,18,23)(11,21)(13,25)(14,26)(15,29,30,37)(17,32,33,38)(19,35,34,39)(20,24)(22,36)(27,31)(28,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,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, 374328);