#135047

order	512
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1802
prime radical	C₂×#135	Q₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^c=eh,h^c=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, 42243);