#133104

order	512
nilpotency class	4
isoclinism	isoclinic class of #133096
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

717 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,21,26,22,25,36,42)(19,27,29,43,30,23,37,39)(20,31,33,44,34,24,35,41)(28,38,45,47,32,40,46,48);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19,22,30)(18,23,25,27)(20,32,34,28)(21,29,36,37)(24,40,31,38)(26,43,42,39)(33,46,35,45)(41,47,44,48);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(17,20,22,34)(18,24,25,31)(19,28,30,32)(21,35,36,33)(23,38,27,40)(26,44,42,41)(29,46,37,45)(39,48,43,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,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, 40300);