#140448

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
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

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

normal subgroups only

This group has 210 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=eh,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,f^b=fi,g^b=gi

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, 47644);