#505584

order	512
nilpotency class	4
isoclinism	isoclinic class of #500112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

597 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,c^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412780);