#484228

order	512
nilpotency class	4
isoclinism	isoclinic class of #484227
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1085 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1085 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)(4,9)(5,21)(6,27)(7,12)(8,15)(10,26)(11,24)(14,39)(16,47)(17,40)(18,32)(19,49)(20,52)(22,37)(23,56)(25,38)(28,35)(29,42)(30,46)(31,55)(33,51)(34,48)(36,44)(41,63)(43,64)(45,58)(50,62)(53,61)(54,59)(57,60);(1,3,7,17)(2,8,12,30)(4,14,18,41)(5,22,25,55)(6,28,24,16)(9,29,32,58)(10,35,36,47)(11,37,27,31)(13,38,40,21)(15,44,46,26)(19,50,23,54)(20,53,48,43)(33,61,60,64)(34,62,52,59)(39,56,63,49)(42,57,45,51);(1,4)(2,9)(3,14)(5,23)(6,20)(7,18)(8,29)(10,33)(11,34)(12,32)(13,39)(15,45)(16,43)(17,41)(19,25)(21,49)(22,54)(24,48)(26,57)(27,52)(28,53)(30,58)(31,59)(35,61)(36,60)(37,62)(38,56)(40,63)(42,46)(44,51)(47,64)(50,55);(4,18)(9,32)(14,41)(19,23)(20,48)(29,58)(33,60)(34,52)(39,63)(42,45)(43,53)(49,56)(50,54)(51,57)(59,62)(61,64)

Polycyclic presentation:

2^9|b^2=i,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^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, 391424);