#500595

order	512
nilpotency class	4
isoclinism	isoclinic class of #500593
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24152
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

769 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283166
maximal normal subgroup of	#72283166

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407791);