#488861

order	512
nilpotency class	4
isoclinism	isoclinic class of #488723
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

627 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396057);