#142119

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = 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

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

normal subgroups only

This group has 292 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eh,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, 49315);