#430657

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(6,9)(7,11)(13,14,20,38,19,25,18,24)(15,26,37,21,30,41,29,40)(16,31,28,42,33,22,32,39)(17,34,27,44,36,43,35,23);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,17)(18,37)(19,30)(20,29)(22,23)(24,41)(25,40)(26,38)(27,32)(28,35)(31,44)(33,36)(34,42)(39,43);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,32,19,33,20,28)(14,22,24,42,25,31,38,39)(15,27,29,17,30,35,37,36)(21,34,26,23,40,43,41,44);(6,9)(10,12)(13,17,18,35,19,36,20,27)(14,23,24,43,25,44,38,34)(15,28,29,16,30,32,37,33)(21,39,26,22,40,42,41,31)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=e,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337853);