#237113

order	512
nilpotency class	3
isoclinism	isoclinic class of #237079
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	#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1223 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=hi,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=di,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, 144309);