#141264

order	512
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=eg,d^2=gh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48460);