#237018

order	512
nilpotency class	3
isoclinism	isoclinic class of #237007
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1211 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:

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

Polycyclic presentation:

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

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, 144214);