#121157

order	512
nilpotency class	3
isoclinism	isoclinic class of #121157
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

363 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,10,12,6)(4,7)(5,8)(9,13)(11,15,16,14)(17,18)(19,29)(20,33,37,25)(21,27)(22,28)(23,26)(24,31)(30,41,45,46)(32,44)(34,39,47,48)(35,42,36,43)(38,40);(1,3,4,11,5,12,13,16)(2,6,7,14,8,10,9,15)(17,19,23,31,22,32,21,38)(18,24,26,44,28,40,27,29)(20,34,35,30,37,47,36,45)(25,41,43,48,33,46,42,39);(2,7,8,9)(6,14,10,15)(17,20,21,35,22,37,23,36)(18,25,26,42,28,33,27,43)(19,30,31,34,32,45,38,47)(24,39,29,41,40,48,44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,e^c=ei,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, 28353);