#508649

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,22)(19,24,29,37)(20,25,32,38)(21,26,36,35)(27,39,44,43)(28,33,41,46)(30,40,47,45)(31,34,42,48);(2,5)(6,8)(17,19,23,29)(18,24,22,37)(20,30,32,47)(21,33,36,46)(25,40,38,45)(26,41,35,28)(27,42,44,31)(34,39,48,43);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,23,32)(18,25,22,38)(19,27,29,44)(21,34,36,48)(24,39,37,43)(26,42,35,31)(28,45,41,40)(30,46,47,33);(3,7)(6,8)(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415845);