#236366

order	512
nilpotency class	3
isoclinism	isoclinic class of #236355
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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,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=dh,e^c=ei,f^c=fhi

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, 143562);