#142160

order	512
nilpotency class	4
isoclinism	isoclinic class of #125076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

416 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,7)(4,5)(6,8)(9,10,12,15,13,16,11,14)(17,18,22,27,30,44,36,26,21,25,37,43,23,28,20,35)(19,29,33,24,39,41,31,46,32,47,48,40,34,45,38,42);(2,5)(6,7)(10,14,16,15)(17,19,30,34,21,32,23,39)(18,24,28,46,25,40,44,42)(20,33,22,38,36,48,37,31)(26,45,27,29,35,41,43,47);(1,3,4,8)(2,6,5,7)(18,25)(24,40)(26,35)(27,43)(28,44)(29,47)(41,45)(42,46)

Polycyclic presentation:

2^9|a^2=f,b^2=defh,c^2=ei,d^2=ghi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49356);