#136565

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1862
prime radical	#466	Q₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

249 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,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,d^b=dg,e^b=egi,d^c=dgi,e^c=eh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43761);