#134818

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,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, 42014);