#240909

order	512
nilpotency class	3
isoclinism	isoclinic class of #240904
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

831 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,31)(25,38)(30,45)(32,40)(33,41)(35,48)(36,44)(37,43)(39,46)(42,47);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(17,19)(18,24)(20,35)(21,33)(22,31)(23,32)(25,42)(26,29)(27,40)(28,41)(30,36)(34,45)(37,47)(38,46)(39,44)(43,48);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,20,21,37,23,38,22,36)(18,25,27,43,28,34,26,44)(19,30,31,46,32,47,33,35)(24,39,29,45,41,48,40,42);(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,36,38,37)(24,40,41,29)(25,43,34,44)(30,35,47,46)(39,45,48,42)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=h,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,g^b=gh,d^c=dh,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148105);