#120976

order	512
nilpotency class	3
isoclinism	isoclinic class of #120974
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

390 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fh,e^c=ehi,g^c=gh

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, 28172);