#203111

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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

686 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,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, 110307);