#466960

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

908 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,24)(20,33)(21,26)(22,36)(23,27)(25,34)(28,31)(29,43)(30,39)(32,40)(35,41)(37,42)(38,44)(45,47)(46,48);(2,5)(6,8)(10,13)(14,16)(17,19,21,30,23,32,22,31)(18,24,26,39,27,40,36,28)(20,29,34,44,35,45,37,46)(25,38,41,47,42,48,33,43);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,18)(19,28)(20,25)(21,36)(22,26)(23,27)(24,31)(29,43)(30,40)(32,39)(33,34)(35,42)(37,41)(38,46)(44,48)(45,47);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20,23,35)(18,25,27,42)(19,29,32,45)(21,34,22,37)(24,38,40,48)(26,41,36,33)(28,43,39,47)(30,44,31,46)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374156);