#513216

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	#33849	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

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

normal subgroups only

This group has 354 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)(13,31,34,65)(14,35,37,59)(16,38,39,24)(18,44,46,33)(19,47,49,29)(23,52,26,51)(25,57,53,70)(28,60,63,64)(32,48,66,69)(36,45,55,72)(40,54,42,56)(41,61,68,67)(43,62,58,71);(1,3)(2,5)(4,8)(6,7)(11,22)(16,39)(21,50)(23,26)(24,38)(25,53)(32,66)(36,55)(40,42)(41,68)(45,72)(48,69)(51,52)(54,56)(57,70)(61,67);(9,11)(10,16)(12,23)(13,32)(14,36)(15,22)(17,40)(18,45)(19,48)(20,39)(21,31)(24,33)(25,29)(26,30)(27,42)(28,61)(34,66)(35,41)(37,55)(38,44)(43,70)(46,72)(47,53)(49,69)(50,65)(51,64)(52,60)(54,62)(56,71)(57,58)(59,68)(63,67);(3,8)(5,7)(9,12,15,30)(10,17,20,27)(11,23,22,26)(13,28,34,63)(14,29,37,47)(16,40,39,42)(18,43,46,58)(19,35,49,59)(21,51,50,52)(24,54,38,56)(25,55,53,36)(31,64,65,60)(32,61,66,67)(33,62,44,71)(41,69,68,48)(45,70,72,57)

Polycyclic presentation:

2^9|a^2=ei,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420412);