#513303

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	C₂×D₃₂	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 326 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=e,d^2=ei,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^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, 420499);