#186209

order	512
nilpotency class	3
isoclinism	isoclinic class of #186137
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62805	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2164 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18,22,28,23,27,21,26)(19,29,31,42,32,24,37,39)(20,33,34,25,36,40,35,41)(30,44,46,38,47,48,45,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,32)(18,24,27,29)(20,30,36,47)(21,31,22,37)(25,38,41,43)(26,39,28,42)(33,48,40,44)(34,45,35,46);(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,40)(27,41)(28,33)(29,43)(31,45)(32,47)(37,46)(39,44)(42,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,27,28)(19,31,32,37)(20,34,36,35)(24,39,29,42)(25,40,41,33)(30,45,47,46)(38,44,43,48)

Polycyclic presentation:

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

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, 93405);