#508476

order	512
nilpotency class	4
isoclinism	isoclinic class of #508310
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

439 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18,22,27,41,54,73,68,21,26,40,53,23,28,43,56)(19,29,32,59,33,49,65,79,31,24,45,47,34,48,66,78)(20,35,38,57,44,75,74,51,37,69,72,52,39,55,42,25)(30,60,62,77,36,70,71,46,61,76,80,50,63,67,64,58);(2,6)(7,13)(8,14)(15,16)(17,19)(18,24)(20,36)(21,31)(22,32)(23,34)(25,50)(26,29)(27,47)(28,49)(30,39)(33,41)(35,67)(37,63)(38,71)(40,45)(42,62)(43,66)(44,61)(46,52)(48,54)(51,77)(53,59)(55,76)(56,79)(57,58)(60,75)(64,72)(65,73)(68,78)(69,70)(74,80);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20,21,37)(18,25,26,51)(19,30,31,61)(22,42,40,74)(23,44,41,39)(24,46,29,58)(27,55,53,75)(28,57,54,52)(32,64,45,71)(33,63,34,36)(35,68,69,56)(38,73,72,43)(47,70,59,67)(48,77,49,50)(60,79,76,78)(62,65,80,66);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(20,37)(25,51)(30,61)(35,69)(36,63)(38,72)(39,44)(42,74)(46,58)(50,77)(52,57)(55,75)(60,76)(62,80)(64,71)(67,70)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415672);