#364626

order	512
nilpotency class	3
isoclinism	isoclinic class of #364625
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13967
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271822);