#508129

order	512
nilpotency class	4
isoclinism	isoclinic class of #508024
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 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 703 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) = 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)(17,18,21,26)(19,29,20,24)(22,36,34,40)(23,39,35,28)(25,30,43,32)(27,46,45,38)(31,42,33,47)(37,41,48,44);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,20)(18,24,26,29)(22,32,34,30)(23,31,35,33)(25,36,43,40)(27,44,45,41)(28,42,39,47)(37,46,48,38);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,21,19)(18,25,26,43)(22,37,34,48)(23,31,35,33)(24,40,29,36)(27,42,45,47)(28,41,39,44)(30,38,32,46);(4,12)(7,14)(10,16)(13,15)(19,20)(24,29)(25,43)(30,32)(31,33)(37,48)(41,44)(42,47)

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,d^b=di,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, 415325);