#142023

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
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

404 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 114 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)(11,13,15,16)(12,14)(17,18,22,27,21,25,23,37)(19,28,32,44,31,42,33,48)(20,34,35,41,29,45,36,26)(24,38,40,46,39,47,43,30);(2,5)(6,7)(9,11,12,15)(10,13)(14,16)(17,19,29,46,22,32,36,47,21,31,20,30,23,33,35,38)(18,24,26,28,37,43,45,48,25,39,41,42,27,40,34,44);(1,3,4,8)(2,6,5,7)(18,25)(24,39)(26,41)(27,37)(28,42)(34,45)(40,43)(44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=defg,c^2=ei,d^2=ghi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49219);