#441263

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

799 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)(19,29)(20,34)(21,24)(22,28)(23,27)(25,32)(26,36)(30,44)(31,41)(33,38)(35,42)(37,39)(40,45)(43,46)(47,48);(1,2,4,5)(3,7,8,6)(9,11,12,15)(10,13,14,16)(18,24)(20,35)(25,38)(27,28)(29,41)(30,45)(36,37)(46,47);(11,15)(13,16)(17,19,23,33,21,31,22,32)(18,25,28,41,24,38,27,29)(20,30,37,47,35,45,36,46)(26,40,42,48,39,44,34,43);(1,3,4,8)(2,6,5,7)(17,20,21,35)(18,26,24,39)(19,30,31,45)(22,36,23,37)(25,40,38,44)(27,34,28,42)(29,43,41,48)(32,46,33,47)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=fi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,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, 348459);