#198683

order	512
nilpotency class	3
isoclinism	isoclinic class of #198528
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1027 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)(9,14)(11,12)(13,15)(17,18,21,26,22,25,27,40)(19,23,30,37,29,38,34,41)(20,31,33,42,32,24,35,39)(28,36,44,47,43,46,45,48);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13)(17,19,27,30,22,29,21,34)(18,23,26,41,25,38,40,37)(20,28,33,45,32,43,35,44)(24,36,42,47,31,46,39,48);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(14,15)(17,20,22,32)(18,24,25,31)(19,28,29,43)(21,35,27,33)(23,36,38,46)(26,42,40,39)(30,45,34,44)(37,48,41,47);(18,25)(19,29)(24,31)(26,40)(28,43)(30,34)(39,42)(44,45)

Polycyclic presentation:

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

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, 105879);