#200126

order	512
nilpotency class	3
isoclinism	isoclinic class of #200122
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

973 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)(4,16,18,9)(5,10)(6,20)(7,11)(8,14)(13,27,30,32)(15,23)(17,26,21,25)(19,24)(22,29,31,28)(33,34)(35,39,44,53)(36,46,47,58)(37,42)(38,41)(40,48,55,50)(43,59)(45,54,49,57)(51,56)(52,62)(60,63)(61,64);(1,3,6,14,7,15,5,19)(2,8,10,12,11,24,20,23)(4,13,21,28,18,30,17,29)(9,22,25,27,16,31,26,32)(33,35)(34,39,41,53)(36,43)(37,49)(38,44)(40,52,55,64)(42,57,56,54)(45,51)(46,63,58,59)(47,60)(48,62)(50,61);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,16)(12,27)(14,28)(15,30)(19,29)(20,26)(23,32)(24,31)(33,36)(34,40)(35,43)(37,50)(38,47)(39,52)(41,55)(42,58)(44,60)(45,62)(46,56)(48,51)(49,61)(53,64)(54,63)(57,59);(1,5,7,6)(2,10,11,20)(3,14,15,19)(4,17,18,21)(8,23,24,12)(9,25,16,26)(13,28,30,29)(22,32,31,27)(35,44)(39,53)(43,60)(45,49)(52,64)(54,57)(59,63)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,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, 107322);