#200605

order	512
nilpotency class	3
isoclinism	isoclinic class of #200430
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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

1111 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18,21,26,22,25,27,40)(19,23,30,38,29,37,34,41)(20,31,33,42,32,24,35,39)(28,36,44,47,43,46,45,48);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,27,30,22,29,21,34)(18,23,26,41,25,37,40,38)(20,28,33,45,32,43,35,44)(24,36,42,47,31,46,39,48);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,20,22,32)(18,24,25,31)(19,28,29,43)(21,35,27,33)(23,36,37,46)(26,42,40,39)(30,45,34,44)(38,48,41,47);(3,10)(8,15)(11,14)(12,16)(18,25)(20,32)(23,37)(26,40)(28,43)(33,35)(38,41)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107801);