#237364

order	512
nilpotency class	3
isoclinism	isoclinic class of #237345
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
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

1232 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,g^c=gi

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, 144560);