#458069

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16309
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 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)(5,25)(6,14)(7,12)(8,13)(9,18)(10,22)(11,26)(16,41)(17,45)(19,32)(21,52)(23,35)(24,36)(27,39)(28,38)(29,37)(30,55)(31,48)(33,44)(34,54)(40,49)(42,59)(43,50)(46,58)(47,57)(51,53)(56,60)(61,63)(62,64);(1,3,6,28,8,18,7,19)(2,9,12,38,13,15,14,32)(4,16,22,55,24,43,23,44)(5,17,26,49,27,47,29,48)(10,33,20,50,35,30,36,41)(11,31,25,57,37,40,39,45)(21,42,53,63,54,46,56,62)(34,59,51,64,52,58,60,61);(1,4,6,22,8,24,7,23)(2,10,12,20,13,35,14,36)(3,16,28,55,18,43,19,44)(5,21,26,53,27,54,29,56)(9,30,38,41,15,33,32,50)(11,34,25,51,37,52,39,60)(17,46,49,62,47,42,48,63)(31,59,57,64,40,58,45,61);(1,5)(2,11)(3,17)(4,21)(6,26)(7,29)(8,27)(9,31)(10,34)(12,25)(13,37)(14,39)(15,40)(16,42)(18,47)(19,48)(20,51)(22,53)(23,56)(24,54)(28,49)(30,58)(32,45)(33,59)(35,52)(36,60)(38,57)(41,61)(43,46)(44,62)(50,64)(55,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365265);