#504180

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

560 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411376);