#479232

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,e^d=ei,h^d=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, 386428);