#203108

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
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	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 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:

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

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,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110304);