#512967

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
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×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 138 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)(11,17)(12,19)(13,30)(14,36)(15,39)(16,22)(18,24)(20,33)(21,32)(23,40)(25,54)(26,44)(27,43)(28,58)(29,51)(31,41)(34,66)(35,62)(37,42)(38,68)(45,52)(46,59)(47,56)(48,53)(49,60)(50,65)(55,69)(57,72)(61,64)(63,71)(67,70);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,27)(13,24)(14,25)(15,23)(16,26)(18,30)(19,43)(20,41)(21,42)(22,44)(28,56)(29,57)(31,33)(32,37)(34,52)(35,53)(36,54)(38,55)(39,40)(45,66)(46,61)(47,58)(48,62)(49,70)(50,71)(51,72)(59,64)(60,67)(63,65)(68,69);(9,11)(10,18)(12,23)(13,31)(14,37)(15,27)(16,26)(17,30)(19,45)(20,42)(21,41)(22,48)(24,33)(25,32)(28,59)(29,55)(34,67)(35,63)(36,46)(38,57)(39,47)(40,58)(43,66)(44,62)(49,69)(50,72)(51,71)(52,60)(53,65)(54,61)(56,64)(68,70);(9,12,16,15)(10,19,22,39)(11,23,26,27)(13,28,35,34)(14,29,38,32)(17,40,44,43)(18,45,48,47)(20,49,50,61)(21,36,51,68)(24,52,53,56)(25,37,55,57)(30,58,62,66)(31,59,63,67)(33,60,65,64)(41,46,71,70)(42,69,72,54)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420163);