#200282

order	512
nilpotency class	3
isoclinism	isoclinic class of #200140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

983 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)(3,12)(4,17,19,9)(5,10)(6,20)(7,11)(8,15)(13,26,29,32)(14,23)(16,27)(18,25,21,24)(22,28,31,30)(33,34,38,44,39,42,37,43)(35,40,48,56,46,54,47,55)(36,49,52,41,50,58,51,57)(45,59,61,53,62,63,60,64);(1,3,6,15,7,16,5,14)(2,8,10,12,11,23,20,27)(4,13,21,30,19,29,18,28)(9,22,24,26,17,31,25,32)(33,35,38,47,39,46,37,48)(34,40,43,56,42,54,44,55)(36,45,51,61,50,62,52,60)(41,53,58,59,57,64,49,63);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,17)(12,26)(14,28)(15,30)(16,29)(20,25)(23,31)(27,32)(33,36)(34,41)(35,45)(37,52)(38,51)(39,50)(40,53)(42,57)(43,58)(44,49)(46,62)(47,61)(48,60)(54,64)(55,63)(56,59);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,19,21)(8,12,23,27)(9,24,17,25)(13,28,29,30)(22,26,31,32)(34,42)(40,54)(41,57)(43,44)(49,58)(53,64)(55,56)(59,63)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107478);