#513084

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
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	#13697	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,25)(13,31)(14,35)(15,37)(16,21)(18,29)(19,27)(20,32)(22,45)(23,49)(24,44)(26,50)(28,51)(30,61)(33,64)(34,63)(36,65)(38,47)(39,46)(40,55)(41,59)(42,68)(43,58)(48,70)(52,57)(53,56)(54,62)(60,71)(66,72)(67,69);(2,5)(6,8)(9,11,16,15)(10,17,21,37)(12,26,28,22)(13,23,34,33)(14,24,36,32)(18,40,41,38)(19,39,43,62)(20,35,44,65)(25,50,51,45)(27,46,58,54)(29,55,59,47)(30,56,60,48)(31,49,63,64)(42,66,67,52)(53,71,70,61)(57,68,72,69);(9,12,27,57,14,30,23,47,11,22,46,69,24,48,34,59,16,28,58,72,36,60,33,55,15,26,54,68,32,56,13,29)(10,18,31,53,20,42,62,50,37,40,64,71,65,66,43,51,21,41,63,70,44,67,39,45,17,38,49,61,35,52,19,25);(1,3,4,7)(2,6,5,8)(12,28)(18,41)(22,26)(25,51)(29,59)(30,60)(38,40)(42,67)(45,50)(47,55)(48,56)(52,66)(53,70)(57,72)(61,71)(68,69)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,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,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232016, #72232017, #72232023, …
maximal normal subgroup of	#72232016, #72232017, #72232023, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420280);