#437662

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

903 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(2,5)(3,7)(9,10,12,14)(11,13,15,16)(17,18,21,27,22,25,23,37)(19,28,31,40,29,41,36,24)(20,32,34,26,35,44,38,42)(30,46,33,48,45,39,47,43);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,24,25,40)(20,33,35,47)(21,36,23,31)(26,43,42,48)(27,41,37,28)(30,38,45,34)(32,46,44,39);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,25)(19,29)(26,42)(27,37)(30,45)(31,36)(32,44)(33,47);(2,5)(6,8)(17,20,23,38,22,35,21,34)(18,26,37,32,25,42,27,44)(19,30,36,47,29,45,31,33)(24,39,41,48,40,46,28,43)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=fi,d^2=ef,e^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,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, 344858);