#193169

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

813 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,22,26,23,27,21,37)(19,28,31,24,32,39,30,40)(20,33,35,25,36,41,34,42)(29,44,46,43,47,38,45,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,31,23,32,21,30)(18,24,26,39,27,40,37,28)(20,29,35,46,36,47,34,45)(25,38,41,48,42,44,33,43);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,38)(26,41)(27,42)(28,43)(30,45)(31,46)(32,47)(33,37)(39,48)(40,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,39,40,28)(25,41,42,33)(29,45,47,46)(38,48,44,43)

Polycyclic presentation:

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

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, 100365);