#121118

order	512
nilpotency class	3
isoclinism	isoclinic class of #121117
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

369 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,g^2=h,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=fh,g^b=ghi,e^c=eh,f^c=fi,g^c=gh

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, 28314);