#508906

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

434 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,18,21)(12,16,20,22)(17,23,19,24)(25,26,31,43,42,51,41,33,30,34,28,35,29,36,45,50)(27,32,44,54,53,55,37,49,38,48,39,47,40,46,52,56);(1,3,4,7)(2,5,6,8)(10,14)(15,21)(16,22)(23,24)(25,27,28,39,42,53,45,52,30,38,31,44,29,40,41,37)(26,32,35,47,51,55,50,56,34,48,43,54,36,46,33,49);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(26,33)(27,37)(28,41)(29,42)(31,45)(32,46)(34,50)(35,36)(38,52)(39,40)(43,51)(44,53)(47,54)(48,55);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(26,34)(27,38)(33,50)(35,43)(36,51)(37,52)(39,44)(40,53)

Polycyclic presentation:

2^9|a^2=egi,b^2=f,c^2=e,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=de,f^c=fh,g^c=gh,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, 416102);