#236408

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,13)(4,18)(5,11)(6,10)(7,12)(8,15)(9,20)(14,25)(16,24)(17,23)(19,27)(21,26)(22,29)(28,30)(31,32)(33,34,38,42)(35,43,45,58)(36,40,48,57)(37,49,52,41)(39,50,54,46)(44,60,61,64)(47,59,51,56)(53,62,55,63);(1,3,7,17)(2,8,12,24)(4,14,19,29)(5,16,6,15)(9,25,26,22)(10,23,11,13)(18,28,27,32)(20,31,21,30)(33,35,38,45)(34,39,42,54)(36,44,48,61)(37,50,52,46)(40,55,57,53)(41,58,49,43)(47,63,51,62)(56,64,59,60);(1,4,7,19)(2,9,12,26)(3,14,17,29)(5,20,6,21)(8,22,24,25)(10,18,11,27)(13,28,23,32)(15,30,16,31)(33,36,38,48)(34,40,42,57)(35,44,45,61)(37,51,52,47)(39,53,54,55)(41,59,49,56)(43,60,58,64)(46,63,50,62);(4,19)(9,26)(14,29)(18,27)(20,21)(22,25)(28,32)(30,31)(35,45)(39,54)(43,58)(44,61)(46,50)(53,55)(60,64)(62,63)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^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=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, 143604);