#493925

order	512
nilpotency class	4
isoclinism	isoclinic class of #481288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#418
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1824
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

979 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)(3,13,16,8)(4,18)(5,22)(6,28)(7,10)(9,20)(11,26)(12,25)(14,41,42,60)(15,45,23,55)(17,32,47,40)(19,50)(21,53)(24,38)(27,54)(29,37)(30,43,39,52)(31,48,36,57)(33,51)(34,58)(35,59)(44,62,49,64)(46,61,56,63);(1,3)(2,8)(4,14)(5,23)(6,17)(7,16)(9,30)(10,13)(11,36)(12,32)(15,27)(18,41)(19,42)(20,52)(21,44)(22,45)(24,56)(25,47)(26,57)(28,40)(29,49)(31,35)(33,39)(34,61)(37,64)(38,63)(43,51)(46,58)(48,59)(50,60)(53,62)(54,55);(1,4,6,29,7,19,25,21)(2,9,28,34,10,33,12,38)(3,14,17,49,16,42,47,44)(5,24,26,20,27,58,59,51)(8,30,40,61,13,39,32,63)(11,37,54,50,35,53,22,18)(15,46,48,43,23,56,57,52)(31,62,45,41,36,64,55,60);(2,10)(8,13)(9,33)(11,35)(12,28)(18,50)(22,54)(30,39)(31,36)(32,40)(34,38)(37,53)(41,60)(45,55)(61,63)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401121);