#500592

order	512
nilpotency class	4
isoclinism	isoclinic class of #500592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24152
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

787 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)(3,16)(4,23)(5,14)(6,32)(7,33)(8,13)(9,15)(10,19)(11,25)(12,30)(17,50)(18,44)(20,48)(21,38)(22,49)(24,59)(26,45)(27,58)(28,40)(29,36)(31,46)(34,41)(35,60)(37,55)(39,51)(42,61)(43,62)(47,52)(53,64)(54,57)(56,63);(1,3,6,19,9,22,8,21)(2,10,13,16,15,38,32,49)(4,24,27,17,28,53,26,54)(5,29,30,55,31,18,7,20)(11,39,42,63,43,47,41,35)(12,44,14,37,33,36,46,48)(23,57,45,64,40,50,58,59)(25,60,34,52,62,56,61,51);(1,4)(2,11)(3,17)(5,25)(6,26)(7,34)(8,27)(9,28)(10,35)(12,40)(13,41)(14,45)(15,43)(16,47)(18,51)(19,24)(20,56)(21,53)(22,54)(23,33)(29,52)(30,61)(31,62)(32,42)(36,59)(37,57)(38,63)(39,49)(44,64)(46,58)(48,50)(55,60);(1,5,6,30,9,31,8,7)(2,12,13,14,15,33,32,46)(3,18,19,20,22,29,21,55)(4,25,26,61,28,62,27,34)(10,36,16,48,38,44,49,37)(11,40,41,45,43,23,42,58)(17,51,24,56,54,52,53,60)(35,59,47,50,63,64,39,57)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307787, #72307945, #72307958
maximal normal subgroup of	#72307787, #72307945, #72307958

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407788);