#121221

order	512
nilpotency class	3
isoclinism	isoclinic class of #117993
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1443
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

353 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,e^2=h,f^2=hi,g^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,f^b=fhi,g^b=ghi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28417);