#134806

order	512
nilpotency class	4
isoclinism	isoclinic class of #134737
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1799
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#380
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 295 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) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42002);