#120519

order	512
nilpotency class	3
isoclinism	isoclinic class of #120516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

431 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27715);