#513218

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
prime radical	#33853	C₂
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

359 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=i,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, 420414);