#508127

order	512
nilpotency class	4
isoclinism	isoclinic class of #508022
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

740 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415323);