#461813

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

783 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)(11,21)(12,25)(13,20)(14,17)(15,19)(16,23)(18,27)(22,31)(24,34)(26,32)(28,33)(29,30)(35,36)(37,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,29,14,24,13,23)(10,16,20,34,17,30,19,21)(12,22,26,36,28,37,27,38)(18,31,33,40,32,39,25,35);(3,7)(5,8)(10,17)(12,26,28,27)(16,30)(18,25,32,33)(19,20)(21,34)(22,36,37,38)(31,35,39,40);(9,12)(10,18)(11,22)(13,27)(14,28)(15,26)(16,31)(17,32)(19,25)(20,33)(21,35)(23,38)(24,37)(29,36)(30,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^d=ei,f^d=fi,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, 369009);