#238416

order	512
nilpotency class	3
isoclinism	isoclinic class of #238414
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

800 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)(3,6)(4,7)(5,9)(8,13)(10,15)(11,14)(12,16)(17,18)(19,29)(20,33,36,43)(21,26)(22,27)(23,28)(24,32)(25,35,41,38)(30,45,34,48)(31,42)(37,40)(39,46,44,47);(1,3)(2,6)(4,12)(5,10)(7,16)(8,14)(9,15)(11,13)(17,19,22,32)(18,24,27,29)(20,34)(21,37,23,31)(25,39)(26,42,28,40)(30,36)(33,48)(35,46)(38,47)(41,44)(43,45);(2,7)(4,13)(6,14)(8,9)(11,12)(15,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,38)(24,39)(26,33)(27,41)(28,43)(29,44)(31,46)(32,34)(37,47)(40,45)(42,48);(3,10)(6,15)(11,12)(14,16)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,35,36,38)(24,40,29,42)(25,33,41,43)(30,46,34,47)(39,45,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145612);