#201578

order	512
nilpotency class	3
isoclinism	isoclinic class of #201414
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×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1097 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18,22,27,23,26,21,38)(19,28,32,40,31,45,30,24)(20,33,36,43,37,25,35,42)(29,44,47,39,34,48,46,41);(1,3,5,11,6,13,4,12)(2,7,8,10,9,16,14,15)(17,19)(18,24,26,40)(20,34)(21,30)(22,32)(23,31)(25,41,33,39)(27,28,38,45)(29,37)(35,47)(36,46)(42,44,43,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,23,37)(18,25,26,33)(19,29,31,34)(21,36,22,35)(24,39,40,41)(27,43,38,42)(28,44,45,48)(30,47,32,46);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(18,26)(24,40)(25,33)(27,38)(28,45)(39,41)(42,43)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^b=fi,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, 108774);