#200125

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	#60953	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

982 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,14)(9,20)(12,25,27,32)(13,26)(16,23,19,24)(21,28,31,30)(22,29)(33,34)(35,40,47,56)(36,49,51,58)(37,42)(38,44)(39,43)(41,50,57,52)(45,59)(46,54,48,55)(53,62)(60,64)(61,63);(1,3,5,14,6,13,20,29)(2,7,9,11,10,22,18,26)(4,12,19,30,17,27,16,28)(8,21,24,25,15,31,23,32)(33,35)(34,40,43,56)(36,45)(37,48)(38,46)(39,47)(41,53,57,64)(42,55,44,54)(49,63,58,59)(50,62)(51,61)(52,60);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,24)(10,15)(11,25)(13,27)(14,30)(16,20)(18,23)(22,31)(26,32)(28,29)(33,36)(34,41)(35,45)(37,50)(38,52)(39,51)(40,53)(42,49)(43,57)(44,58)(46,60)(47,61)(48,62)(54,59)(55,63)(56,64);(3,13)(7,22)(11,26)(12,27)(14,29)(21,31)(25,32)(28,30)(33,37,39,38)(34,42,43,44)(35,46,47,48)(36,50,51,52)(40,54,56,55)(41,49,57,58)(45,60,61,62)(53,59,64,63)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,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, 107321);