#205538

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1018 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18)(19,24,30,39)(20,32)(21,26)(22,36)(23,27)(25,34)(28,41,43,48)(29,38,31,45)(33,40)(35,46)(37,42,47,44);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19)(18,24,27,39)(20,28)(21,29)(22,31)(23,30)(25,37,40,47)(26,38,36,45)(32,41,46,48)(33,42)(34,44)(35,43);(2,7)(6,14)(8,12)(9,15)(17,20,21,33,23,35,22,34)(18,25,36,46,27,40,26,32)(19,28,29,42,30,43,31,44)(24,37,45,48,39,47,38,41);(17,21,23,22)(18,26,27,36)(19,29,30,31)(20,33,35,34)(24,38,39,45)(25,32,40,46)(28,42,43,44)(37,41,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112734);