#485596

order	512
nilpotency class	4
isoclinism	isoclinic class of #485592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18152
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,15)(4,20)(5,24)(6,13)(7,11)(8,14)(9,17)(10,23)(12,28)(16,45)(18,43)(19,44)(21,38)(22,55)(25,34)(26,36)(27,39)(29,37)(30,41)(31,40)(32,59)(33,52)(35,51)(42,48)(46,62)(47,53)(49,64)(50,60)(54,58)(56,61)(57,63);(1,3)(2,9)(4,16)(5,25)(6,18)(7,31)(8,19)(10,32)(11,33)(12,40)(13,34)(14,35)(15,24)(17,28)(20,53)(21,46)(22,56)(23,48)(26,59)(27,52)(29,49)(30,51)(36,62)(37,61)(38,60)(39,44)(41,43)(42,54)(45,55)(47,58)(50,57)(63,64);(1,4,8,21)(2,10,14,37)(3,16,19,46)(5,26,27,22)(6,29,7,23)(9,32,35,61)(11,38,13,20)(12,36,41,55)(15,42,44,64)(17,50,51,47)(18,49,31,48)(24,54,39,63)(25,59,52,56)(28,57,30,58)(33,60,34,53)(40,62,43,45);(2,11,14,13)(4,21)(5,27)(9,33,35,34)(10,20,37,38)(12,24,41,39)(15,43,44,40)(16,46)(17,51)(23,29)(25,52)(28,30)(32,53,61,60)(36,63,55,54)(42,62,64,45)(48,49)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392792);