#138218

order	512
nilpotency class	4
isoclinism	isoclinic class of #125524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

382 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

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, 45414);