#152275

order	512
nilpotency class	5
isoclinism	isoclinic class of #150138
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6)(7,17,18,45)(8,12)(9,25)(10,30)(11,33)(13,19,35,50)(14,28)(15,32)(16,24)(20,56,51,68)(21,57,34,49)(22,46)(23,41)(26,43,54,47)(27,37)(29,64)(31,48,53,40)(36,52)(38,65,67,61)(39,66,44,55)(42,62)(58,60)(59,63);(1,2,3,4)(6,12)(13,35)(14,36)(15,37)(16,33)(17,45)(25,46)(30,41)(38,67)(39,44)(40,48)(42,59)(43,47)(49,57)(56,68)(60,64);(5,7)(6,13)(8,18)(9,26)(10,31)(11,34)(12,35)(14,40)(15,43)(16,44)(17,46)(19,52)(20,32)(21,24)(22,54)(23,53)(25,45)(27,51)(28,50)(29,65)(30,38)(33,39)(36,48)(37,47)(41,67)(42,56)(49,60)(55,63)(57,64)(58,61)(59,68)(62,66)

Polycyclic presentation:

2^9|a^2=f,b^2=di,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^c=egi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59471);