#439136

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

994 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:

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

Polycyclic presentation:

2^9|b^2=efg,c^2=f,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi

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, 346332);