#135380

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=d,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^c=eh,f^c=fghi,g^c=gi,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, 42576);