#205563

order	512
nilpotency class	3
isoclinism	isoclinic class of #205561
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112759);