#135649

order	512
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#106
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,6)(4,5)(7,8)(9,10,11,13,12,14,15,16)(17,18)(19,29,33,44)(20,35,34,48)(21,37)(22,38)(23,28)(24,32,42,45)(25,36,43,46)(26,31)(27,39)(30,41)(40,47);(2,5)(6,8)(17,19,30,45,22,34,47,36,23,33,21,32,39,20,31,46)(18,24,40,44,27,25,41,48,28,42,26,29,38,43,37,35);(1,3,4,7)(2,6,5,8)(17,20,21,36,22,19,31,32,23,34,30,46,39,33,47,45)(18,25,26,35,27,42,37,44,28,43,40,48,38,24,41,29)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=d,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,f^b=fi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42845);