#133967

order	512
nilpotency class	4
isoclinism	isoclinic class of #133877
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

720 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41163);