#479283

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

786 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 162 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)(5,6,11,16,12,18,10,17)(7,19,23,39,24,47,22,13)(8,25,28,43,29,42,36,14)(9,30,34,46,35,15,33,45)(20,50,52,48,53,37,26,40)(21,54,57,61,31,49,56,38)(27,44,32,41,60,62,59,65)(51,66,55,64,68,63,58,67);(5,7,10,22,12,24,11,23)(6,13,16,19,18,39,17,47)(8,26,28,20,29,52,36,53)(9,31,33,57,35,21,34,56)(14,40,42,37,43,48,25,50)(15,38,45,54,30,61,46,49)(27,58,32,51,60,55,59,68)(41,64,44,66,65,67,62,63);(5,8,10,28,12,29,11,36)(6,14,16,42,18,43,17,25)(7,20,22,52,24,53,23,26)(9,32,33,60,35,59,34,27)(13,37,19,48,39,50,47,40)(15,44,45,65,30,62,46,41)(21,55,56,68,31,58,57,51)(38,64,54,66,61,67,49,63);(2,4)(5,9,10,34,12,35,11,33)(6,15,17,45,18,30,16,46)(7,21,23,56,24,31,22,57)(8,27,28,59,29,60,36,32)(13,38,19,49,39,61,47,54)(14,41,25,44,43,65,42,62)(20,51,26,55,53,68,52,58)(37,63,48,67,50,66,40,64)

Polycyclic presentation:

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

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, 386479);