#503161

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61091	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1104 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,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=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410357);