#201643

order	512
nilpotency class	3
isoclinism	isoclinic class of #201434
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

1056 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^b=fi,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, 108839);