#140999

order	512
nilpotency class	4
isoclinism	isoclinic class of #125409
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 76 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)(5,6,10,16,11,17,31,32,9,15,18,44,29,14,42,43)(7,12,22,35,20,36,30,60,21,34,53,61,23,46,47,45)(8,24,27,55,28,39,59,66,26,13,37,40,57,41,49,62)(19,48,51,68,25,54,56,67,50,33,58,63,52,38,64,65);(5,7,18,47,29,20,31,22,9,21,10,30,11,23,42,53)(6,12,32,61,17,46,44,60,15,34,43,35,14,36,16,45)(8,25,37,51,57,50,59,56,26,52,27,58,28,19,49,64)(13,38,62,67,41,33,55,68,24,54,66,65,39,48,40,63);(1,2,3,4)(5,8,11,28,9,26,29,57)(6,13,17,41,15,24,14,39)(7,19,20,25,21,50,23,52)(10,27,31,59,18,37,42,49)(12,33,36,38,34,48,46,54)(16,40,32,62,44,55,43,66)(22,51,30,56,53,58,47,64)(35,63,60,65,61,68,45,67)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=egi,d^2=ghi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,f^d=fi

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, 48195);