#121348

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

454 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28544);