#508474

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

455 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=e,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, 415670);