#133156

order	512
nilpotency class	4
isoclinism	isoclinic class of #133109
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,10,12,7)(4,8)(5,9)(6,11,15,13)(14,16)(17,18,22,28,23,29,21,27)(19,30,33,25,34,41,32,42)(20,26,35,43,36,24,37,39)(31,45,46,38,47,44,48,40);(1,3)(2,6)(4,13)(5,12)(7,8)(9,15)(10,16)(11,14)(18,24,29,26)(25,38,42,40)(27,39,28,43)(30,44,41,45);(1,3)(2,7)(4,11)(5,12)(6,8)(9,10)(13,14)(15,16)(17,19,23,34)(18,25,29,42)(20,31,36,47)(21,33,22,32)(24,38,26,40)(27,41,28,30)(35,48,37,46)(39,45,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40352);