#121105

order	512
nilpotency class	3
isoclinism	isoclinic class of #121103
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=gi,f^c=fhi,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, 28301);