#186893

order	512
nilpotency class	3
isoclinism	isoclinic class of #186835
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
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

2151 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:

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

Polycyclic presentation:

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

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, 94089);