#508475

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

Polycyclic presentation:

2^9|a^2=fi,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, 415671);