#203097

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,d^c=dh,f^c=fhi

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, 110293);