#481444

order	512
nilpotency class	4
isoclinism	isoclinic class of #481438
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#327
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60462	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388640);