#121213

order	512
nilpotency class	3
isoclinism	isoclinic class of #121211
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,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=fi,g^b=ghi,e^c=eh,f^c=fh,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, 28409);