#142104

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, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=defh,c^2=ei,d^2=ghi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,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, 49300);