#121232

order	512
nilpotency class	3
isoclinism	isoclinic class of #121229
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 365 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)(17,18,22,27,23,26,21,36)(19,28,31,40,30,24,37,41)(20,32,34,43,35,25,38,44)(29,45,47,48,46,39,33,42);(1,3,4,11,5,12,13,16)(2,6,7,14,8,10,9,15)(17,19)(18,24,26,28)(20,33)(21,37)(22,31)(23,30)(25,42,32,48)(27,41,36,40)(29,34)(35,47)(38,46)(39,44,45,43);(2,7,8,9)(6,14,10,15)(17,20)(18,25)(19,29,30,46)(21,34)(22,38)(23,35)(24,39,28,45)(26,32)(27,43)(31,33,37,47)(36,44)(40,42,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gh,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, 28428);