#456741

order	512
nilpotency class	3
isoclinism	isoclinic class of #455397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16285
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

936 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,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, 363937);