#240708

order	512
nilpotency class	3
isoclinism	isoclinic class of #240703
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 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)(10,14)(11,16)(12,15)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,31)(25,37)(29,43)(30,40)(32,42)(34,48)(35,47)(36,41)(39,45)(44,46);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,39)(26,42)(27,40)(28,38)(29,37)(33,46)(35,44)(36,45)(41,48)(43,47);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,20,22,36,23,37,21,35)(18,25,26,41,27,33,38,47)(19,29,31,45,32,34,30,44)(24,39,28,43,40,46,42,48);(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,37,36)(24,28,40,42)(25,41,33,47)(29,44,34,45)(39,43,46,48)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,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, 147904);