#447750

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

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efg,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, 354946);