#436577

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=e,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 343773);