#508461

order	512
nilpotency class	4
isoclinism	isoclinic class of #508286
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 489 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) = 48
transitive degree	μ_t(G) = 128

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)(17,18,23,25,22,28,20,26)(19,24,32,40,33,41,30,38)(21,27,36,42,35,45,34,43)(29,44,46,37,47,48,31,39);(2,6)(7,13)(8,14)(15,16)(17,19)(18,24)(20,30)(21,29)(22,33)(23,32)(25,40)(26,38)(27,44)(28,41)(31,34)(35,47)(36,46)(37,42)(39,43)(45,48);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(18,25)(19,29)(20,23)(21,34)(24,37)(26,28)(30,46)(31,32)(33,47)(35,36)(38,48)(39,41)(40,44)(42,43);(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=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415657);