#238342

order	512
nilpotency class	3
isoclinism	isoclinic class of #238339
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 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)(3,10,12,16)(4,8)(5,14)(6,9)(7,13,15,11)(17,18)(19,23)(20,30)(21,33)(22,26)(24,31)(25,36)(27,44)(28,38)(29,42)(32,43)(34,39)(35,41)(37,46)(40,47)(45,48);(3,11,12,13)(7,10,15,16)(17,19)(18,23)(20,27)(21,34)(22,28)(24,40)(25,42)(26,38)(29,36)(30,44)(31,47)(32,45)(33,39)(35,46)(37,41)(43,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,16)(9,15)(10,14)(17,20,22,32)(18,24,26,41)(19,27,28,45)(21,35,36,31)(23,37,38,40)(25,43,33,30)(29,47,34,46)(39,48,42,44);(19,28)(23,38)(27,45)(29,34)(37,40)(39,42)(44,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145538);