#513256

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,5,4)(3,7,8,6)(9,10,12,18)(11,22,23,16)(13,17,27,40)(14,33,28,62)(15,34,29,21)(19,45,43,30)(20,47,44,31)(24,35,52,39)(25,59,53,65)(26,38,54,51)(32,48,61,42)(36,57,49,55)(37,56,50,60)(41,64,46,63)(58,68,70,67)(66,71,69,72);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,26)(17,39)(18,22)(19,36)(20,37)(21,38)(27,52)(28,53)(29,54)(30,57)(31,56)(32,58)(33,65)(34,51)(35,40)(41,69)(42,68)(43,49)(44,50)(45,55)(46,66)(47,60)(48,67)(59,62)(61,70)(63,72)(64,71);(1,4,5,2)(3,7,8,6)(10,17)(11,23)(13,30)(14,31)(15,29)(16,35)(18,40)(19,20)(21,48)(22,39)(24,55)(25,60)(27,45)(28,47)(32,64)(33,41)(34,42)(36,50)(37,49)(38,68)(43,44)(46,62)(51,67)(52,57)(53,56)(58,72)(59,69)(61,63)(65,66)(70,71);(3,8)(6,7)(10,18)(11,23)(17,40)(19,43)(20,44)(21,34)(24,52)(25,53)(26,54)(33,62)(41,46)(42,48)(55,57)(56,60)(58,70)(71,72)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260483
maximal normal subgroup of	#72260483

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420452);