#236415

order	512
nilpotency class	3
isoclinism	isoclinic class of #236399
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	#60959	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1215 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,31)(29,44)(30,38)(32,43)(33,40)(34,42)(36,41)(37,47)(39,45)(46,48);(2,7)(4,5)(8,11)(9,10)(13,14)(15,16)(17,19,23,32)(18,24,27,38)(20,29,34,46)(21,31,22,30)(25,39,42,37)(26,43,35,28)(33,47,36,45)(40,44,41,48);(1,3)(2,8)(4,13)(5,14)(6,12)(7,15)(9,11)(10,16)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,40)(27,42)(28,44)(30,45)(31,47)(32,46)(35,41)(38,39)(43,48);(3,12)(8,16)(11,15)(13,14)(17,21,23,22)(18,26,27,35)(19,30,32,31)(20,33,34,36)(24,28,38,43)(25,40,42,41)(29,45,46,47)(37,44,39,48)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=h,f^2=hi,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,d^c=dhi,e^c=ei,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, 143611);