#202113

order	512
nilpotency class	3
isoclinism	isoclinic class of #201958
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1016 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)(3,5)(4,11)(6,9)(7,13)(8,10)(12,16)(14,15)(17,18,22,27,23,26,21,28)(19,29,33,39,31,42,32,24)(20,34,36,25,37,41,35,40)(30,38,46,47,44,48,45,43);(1,3,10,9,6,8,5,2)(4,11,13,16,14,15,12,7)(17,19,23,31)(18,24)(20,30,37,44)(21,32,22,33)(25,38)(26,39)(27,29)(28,42)(34,43)(35,45,36,46)(40,48)(41,47);(1,4,6,14)(2,7,9,16)(3,11,8,15)(5,12,10,13)(17,20,23,37)(18,25,26,40)(19,30,31,44)(21,36,22,35)(24,38,39,48)(27,34,28,41)(29,43,42,47)(32,46,33,45);(1,5,6,10)(2,3,9,8)(4,12,14,13)(7,11,16,15)(18,26)(19,31)(25,40)(27,28)(30,44)(32,33)(34,41)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109309);