#502906

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 803 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10,16,22)(11,17,27,45)(12,28,33,65)(13,34,36,20)(14,19,38,52)(15,39,35,21)(18,46,50,30)(23,56,59,71)(24,51,60,43)(25,42,62,53)(26,55,40,44)(29,63,61,72)(31,49,37,64)(32,54,67,48)(41,68,69,57)(47,58,70,66);(2,5)(6,7)(9,11,16,27)(10,17,22,45)(12,29,33,61)(13,25,36,62)(14,24,38,60)(15,40,35,26)(18,47,50,70)(19,51,52,43)(20,53,34,42)(21,55,39,44)(23,37,59,31)(28,63,65,72)(30,66,46,58)(32,68,67,57)(41,54,69,48)(49,56,64,71);(9,12,14,37,15,32,36,46,16,33,38,31,35,67,13,30)(10,18,34,48,21,49,52,65,22,50,20,54,39,64,19,28)(11,23,25,61,26,58,60,68,27,59,62,29,40,66,24,57)(17,41,51,47,44,63,53,71,45,69,43,70,55,72,42,56);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

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, 410102);