#119361

order	512
nilpotency class	3
isoclinism	isoclinic class of #119357
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

554 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26557);