#484653

order	512
nilpotency class	4
isoclinism	isoclinic class of #484652
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#33530	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

873 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,13)(4,18)(5,22)(6,28)(7,12)(8,15)(9,19)(10,27)(11,25)(14,41)(16,48)(17,40)(20,53)(21,34)(23,37)(24,57)(26,38)(29,35)(30,46)(31,47)(32,59)(33,60)(36,45)(39,42)(43,49)(44,62)(50,55)(51,52)(54,61)(56,64)(58,63);(1,3,7,17)(2,8,12,31)(4,14,19,42)(5,23,26,59)(6,29,25,16)(9,30,18,49)(10,35,36,48)(11,37,28,32)(13,38,40,22)(15,45,47,27)(20,54,24,58)(21,56,52,44)(33,62,50,64)(34,63,51,61)(39,53,41,57)(43,60,46,55);(1,4,7,19)(2,9)(3,14,17,42)(5,24,26,20)(6,21,25,52)(8,30)(10,33)(11,34)(12,18)(13,39)(15,46,47,43)(16,44,29,56)(22,57)(23,58,59,54)(27,60,45,55)(28,51)(31,49)(32,61)(35,62)(36,50)(37,63)(38,53)(40,41)(48,64);(4,19)(9,18)(14,42)(20,24)(21,52)(30,49)(33,50)(34,51)(39,41)(43,46)(44,56)(53,57)(54,58)(55,60)(61,63)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391849);