#121324

order	512
nilpotency class	3
isoclinism	isoclinic class of #121075
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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

371 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,10,13,11)(4,14)(5,8)(6,9)(7,12,15,16)(17,18)(19,27,30,40)(20,24)(21,25)(22,26)(23,29,37,34)(28,43,31,42)(32,38)(33,39)(35,41)(36,46,47,45)(44,48);(1,3,6,13)(2,7)(4,11)(5,12,14,16)(8,10)(9,15)(17,19,21,34,22,30,35,29)(18,23,25,40,26,37,41,27)(20,31,32,45,33,28,44,46)(24,36,38,43,39,47,48,42);(1,4,5,2,6,8,14,9)(3,11,12,7,13,10,16,15)(17,20,21,32,22,33,35,44)(18,24,25,38,26,39,41,48)(19,28,29,45,30,31,34,46)(23,36,27,42,37,47,40,43)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28520);