#133108

order	512
nilpotency class	4
isoclinism	isoclinic class of #133089
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#537	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 703 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)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,21,26,22,25,35,40)(19,27,30,41,29,23,36,38)(20,31,33,42,32,24,34,39)(28,43,45,48,44,37,46,47);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,37)(25,27)(26,41)(31,43)(32,44)(33,45)(34,46)(35,36)(38,40)(39,48)(42,47);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(17,20,22,32)(18,24,25,31)(19,28,29,44)(21,34,35,33)(23,37,27,43)(26,42,40,39)(30,46,36,45)(38,48,41,47)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,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, 40304);