#513023

order	512
nilpotency class	5
isoclinism	isoclinic class of #513022
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#33669
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#3414	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,f^b=fi,d^c=dh,e^c=ei,f^c=fg,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420219);