#237168

order	512
nilpotency class	3
isoclinism	isoclinic class of #237155
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

1216 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)(3,11,14,15)(4,8,5,9)(7,13,16,12)(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);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,11,9,15)(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);(3,12)(4,5)(7,11)(8,9)(13,14)(15,16)(17,20)(18,25)(19,30)(21,37)(22,36)(23,38)(24,39)(26,34)(27,43)(28,44)(29,42)(31,35)(32,45)(33,46)(40,48)(41,47);(1,4,6,5)(2,8,10,9)(3,13,14,12)(7,11,16,15)(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,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=dhi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144364);