#135128

order	512
nilpotency class	4
isoclinism	isoclinic class of #135093
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#466	Q₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

265 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=di,d^2=g,e^2=ghi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^c=egh,f^c=fg,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42324);