#121230

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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 375 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,e^2=h,f^2=i,g^2=h,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, 28426);