#133790

order	512
nilpotency class	4
isoclinism	isoclinic class of #133470
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

524 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40986);