#434679

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^d=ei,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, 341875);