#437120

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

775 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)(6,7)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,33)(28,40)(29,43)(34,38)(36,42)(37,48)(39,41)(44,45)(46,47);(1,2,4,5)(3,7,8,6)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(20,32)(26,35)(30,42)(34,47)(37,45)(39,43);(11,15)(13,16)(17,19,22,30,21,28,33,42)(18,24,26,27,23,36,35,40)(20,29,34,44,32,41,47,48)(25,37,38,39,31,45,46,43);(1,3,4,8)(2,6,5,7)(17,20,33,47,21,32,22,34)(18,25,35,46,23,31,26,38)(19,29,42,48,28,41,30,44)(24,37,40,43,36,45,27,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=ei,c^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,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, 344316);