#240914

order	512
nilpotency class	3
isoclinism	isoclinic class of #240913
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

895 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,16)(14,15)(17,18,23,28)(19,29,31,41)(20,34,37,25)(21,27,22,26)(24,33,40,32)(30,45,46,42)(35,48,47,39)(36,43,38,44);(4,13)(6,15)(7,12)(11,14)(17,19)(18,24)(20,35)(21,33)(22,32)(23,31)(25,42)(26,41)(27,29)(28,40)(30,38)(34,45)(36,46)(37,47)(39,44)(43,48);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,20,21,36,23,37,22,38)(18,25,26,43,28,34,27,44)(19,30,32,47,31,46,33,35)(24,39,29,45,40,48,41,42);(3,10)(6,15)(9,16)(11,14)(19,31)(24,40)(29,41)(30,46)(32,33)(35,47)(39,48)(42,45)

Polycyclic presentation:

2^9|a^2=h,c^2=e,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=di,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, 148110);