#503554

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	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

719 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ei,c^2=efh,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, 410750);