#481439

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

1158 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1158 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)(3,14,17,41)(4,10)(5,24,26,39)(6,12)(7,29)(8,13)(9,16)(11,27,37,28)(15,40,42,61)(18,36,19,48)(20,33)(21,50,49,60)(22,35)(23,54)(25,38)(30,43)(31,46)(32,47)(34,52,57,53)(44,58,45,64)(51,59)(55,62)(56,63);(1,3,8,17)(2,9,13,32)(4,15,20,42)(5,25,26,46)(6,19,7,18)(10,30,33,55)(11,36,37,48)(12,38,29,31)(14,39,41,24)(16,27,47,28)(21,51,49,63)(22,45,23,44)(34,58,57,64)(35,59,54,56)(40,60,61,50)(43,52,62,53);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,20)(9,30)(11,34)(12,35)(13,33)(14,40)(16,43)(17,42)(18,44)(19,45)(24,50)(25,51)(26,49)(27,52)(28,53)(29,54)(31,56)(32,55)(36,58)(37,57)(38,59)(39,60)(41,61)(46,63)(47,62)(48,64);(4,20)(10,33)(15,42)(21,49)(22,23)(30,55)(34,57)(35,54)(40,61)(43,62)(44,45)(50,60)(51,63)(52,53)(56,59)(58,64)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=hi,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, 388635);